formula for experimental

Experimental Probability

The chance or occurrence of a particular event is termed its probability. The value of a probability lies between 0 and 1 which means if it is an impossible event, the probability is 0 and if it is a certain event, the probability is 1. The probability that is determined on the basis of the results of an experiment is known as experimental probability. This is also known as empirical probability.

What is Experimental Probability?

Experimental probability is a probability that is determined on the basis of a series of experiments. A random experiment is done and is repeated many times to determine their likelihood and each repetition is known as a trial. The experiment is conducted to find the chance of an event to occur or not to occur. It can be tossing a coin, rolling a die, or rotating a spinner. In mathematical terms, the probability of an event is equal to the number of times an event occurred ÷ the total number of trials. For instance, you flip a coin 30 times and record whether you get a head or a tail. The experimental probability of obtaining a head is calculated as a fraction of the number of recorded heads and the total number of tosses. P(head) = Number of heads recorded ÷ 30 tosses.

Experimental Probability Formula

The experimental probability of an event is based on the number of times the event has occurred during the experiment and the total number of times the experiment was conducted. Each possible outcome is uncertain and the set of all the possible outcomes is called the sample space. The formula to calculate the experimental probability is: P(E) = Number of times an event occurs/Total number of times the experiment is conducted

Consider an experiment of rotating a spinner 50 times. The table given below shows the results of the experiment conducted. Let us find the experimental probability of spinning the color - blue.

The experimental probability of spinning the color blue = 10/50 = 1/5 = 0.2 = 20%

Experimental Probability vs Theoretical Probability

Experimental results are unpredictable and may not necessarily match the theoretical results. The results of experimental probability are close to theoretical only if the number of trials is more in number. Let us see the difference between experimental probability and theoretical probability.

Experimental Probability Examples

Here are a few examples from real-life scenarios.

a) The number of cookies made by Patrick per day in this week is given as 4, 7, 6, 9, 5, 9, 5.

Based on this data, what is the reasonable estimate of the probability that Patrick makes less than 6 cookies the next day?

P(< 6 cookies) = 3/7 = 0.428 = 42%

b) Find the reasonable estimate of the probability that while ordering a pizza, the next order will not be of a pepperoni topping.

Based on this data , the reasonable estimate of the probability that the next type of toppings that would get ordered is not a pepperoni will be 15/20 = 3/4 = 75%

Related Sections

• Card Probability
• Conditional Probability Calculator
• Binomial Probability Calculator
• Probability Rules
• Probability and Statistics

Important Notes

• The sum of the experimental probabilities of all the outcomes is 1.
• The probability of an event lies between 0 and 1, where 0 is an impossible event and 1 denotes a certain event.
• Probability can also be expressed in percentage.

Examples on Experimental Probability

Example 1: The following table shows the recording of the outcomes on throwing a 6-sided die 100 times.

Find the experimental probability of: a) Rolling a four; b) Rolling a number less than four; c) Rolling a 2 or 5

Experimental probability is calculated by the formula: Number of times an event occurs/Total number of trials

a) Rolling a 4: 17/100 = 0.17

b) Rolling a number less than 4: 56/100 = 0.56

c) Rolling a 2 or 5: 31/100 = 0.31

Example 2: The following set of data shows the number of messages that Mike received recently from 6 of his friends. 4, 3, 2, 1, 6, 8. Based on this, find the probability that Mike will receive less than 2 messages next time.

Mike has received less than 2 messages from 2 of his friends out of 6.

Therefore, P(<2) = 2/6 = 1/3

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Practice Questions on Experimental Probability

Frequently asked questions (faqs), how do you find the experimental probability.

The experimental probability of an event is based on actual experiments and the recordings of the events. It is equal to the number of times an event occurred divided by the total number of trials.

What is the Experimental Probability of rolling a 6?

The experimental probability of rolling a 6 is 1/6. A die has 6 faces numbered from 1 to 6. Rolling the die to get any number from 1 to 6 is the same and the probability (of getting a 6) = Number of favorable outcomes/ total possible outcomes = 1/6.

What is the Difference Between Theoretical and Experimental Probability?

Theoretical probability is what is expected to happen and experimental probability is what has actually happened in the experiment.

Do You Simplify Experimental Probability?

Yes, after finding the ratio of the number of times the event occurred to the total number of trials conducted, the fraction which is obtained is simplified.

Which Probability is More Accurate, Theoretical Probability or Experimental Probability?

Theoretical probability is more accurate than experimental probability. The results of experimental probability are close to theoretical only if the number of trials are more in number.

3.2 Determining Empirical and Molecular Formulas

Learning objectives.

By the end of this section, you will be able to:

• Compute the percent composition of a compound
• Determine the empirical formula of a compound
• Determine the molecular formula of a compound

The previous section discussed the relationship between the bulk mass of a substance and the number of atoms or molecules it contains (moles). Given the chemical formula of the substance, one may determine the amount of the substance (moles) from its mass, and vice versa. But what if the chemical formula of a substance is unknown? In this section, these same principles will be applied to derive the chemical formulas of unknown substances from experimental mass measurements.

Percent Composition

The elemental makeup of a compound defines its chemical identity, and chemical formulas are the most succinct way of representing this elemental makeup. When a compound’s formula is unknown, measuring the mass of each of its constituent elements is often the first step in the process of determining the formula experimentally. The results of these measurements permit the calculation of the compound’s percent composition , defined as the percentage by mass of each element in the compound. For example, consider a gaseous compound composed solely of carbon and hydrogen. The percent composition of this compound could be represented as follows:

If analysis of a 10.0-g sample of this gas showed it to contain 2.5 g H and 7.5 g C, the percent composition would be calculated to be 25% H and 75% C:

Example 3.9

Calculation of percent composition.

The analysis results indicate that the compound is 61.0% C, 15.4% H, and 23.7% N by mass.

Check Your Learning

12.1% C, 16.1% O, 71.79% Cl

Determining Percent Composition from Molecular or Empirical Formulas

Percent composition is also useful for evaluating the relative abundance of a given element in different compounds of known formulas. As one example, consider the common nitrogen-containing fertilizers ammonia (NH 3 ), ammonium nitrate (NH 4 NO 3 ), and urea (CH 4 N 2 O). The element nitrogen is the active ingredient for agricultural purposes, so the mass percentage of nitrogen in the compound is a practical and economic concern for consumers choosing among these fertilizers. For these sorts of applications, the percent composition of a compound is easily derived from its formula mass and the atomic masses of its constituent elements. A molecule of NH 3 contains one N atom weighing 14.01 amu and three H atoms weighing a total of (3 × × 1.008 amu) = 3.024 amu. The formula mass of ammonia is therefore (14.01 amu + 3.024 amu) = 17.03 amu, and its percent composition is:

This same approach may be taken considering a pair of molecules, a dozen molecules, or a mole of molecules, etc. The latter amount is most convenient and would simply involve the use of molar masses instead of atomic and formula masses, as demonstrated Example 3.10 . As long as the molecular or empirical formula of the compound in question is known, the percent composition may be derived from the atomic or molar masses of the compound's elements.

Example 3.10

Determining percent composition from a molecular formula.

Note that these percentages sum to equal 100.00% when appropriately rounded.

Determination of Empirical Formulas

As previously mentioned, the most common approach to determining a compound’s chemical formula is to first measure the masses of its constituent elements. However, keep in mind that chemical formulas represent the relative numbers , not masses, of atoms in the substance. Therefore, any experimentally derived data involving mass must be used to derive the corresponding numbers of atoms in the compound. This is accomplished using molar masses to convert the mass of each element to a number of moles. These molar amounts are used to compute whole-number ratios that can be used to derive the empirical formula of the substance. Consider a sample of compound determined to contain 1.71 g C and 0.287 g H. The corresponding numbers of atoms (in moles) are:

Thus, this compound may be represented by the formula C 0.142 H 0.284 . Per convention, formulas contain whole-number subscripts, which can be achieved by dividing each subscript by the smaller subscript:

(Recall that subscripts of “1” are not written but rather assumed if no other number is present.)

The empirical formula for this compound is thus CH 2 . This may or may not be the compound’s molecular formula as well; however, additional information is needed to make that determination (as discussed later in this section).

Consider as another example a sample of compound determined to contain 5.31 g Cl and 8.40 g O. Following the same approach yields a tentative empirical formula of:

In this case, dividing by the smallest subscript still leaves us with a decimal subscript in the empirical formula. To convert this into a whole number, multiply each of the subscripts by two, retaining the same atom ratio and yielding Cl 2 O 7 as the final empirical formula.

In summary, empirical formulas are derived from experimentally measured element masses by:

• Deriving the number of moles of each element from its mass
• Dividing each element’s molar amount by the smallest molar amount to yield subscripts for a tentative empirical formula
• Multiplying all coefficients by an integer, if necessary, to ensure that the smallest whole-number ratio of subscripts is obtained

Figure 3.11 outlines this procedure in flow chart fashion for a substance containing elements A and X.

Example 3.11

Determining a compound’s empirical formula from the masses of its elements.

Next, derive the iron-to-oxygen molar ratio by dividing by the lesser number of moles:

The ratio is 1.000 mol of iron to 1.500 mol of oxygen (Fe 1 O 1.5 ). Finally, multiply the ratio by two to get the smallest possible whole number subscripts while still maintaining the correct iron-to-oxygen ratio:

The empirical formula is Fe 2 O 3 .

Link to Learning

For additional worked examples illustrating the derivation of empirical formulas, watch the brief video clip.

Deriving Empirical Formulas from Percent Composition

Finally, with regard to deriving empirical formulas, consider instances in which a compound’s percent composition is available rather than the absolute masses of the compound’s constituent elements. In such cases, the percent composition can be used to calculate the masses of elements present in any convenient mass of compound; these masses can then be used to derive the empirical formula in the usual fashion.

Example 3.12

Determining an empirical formula from percent composition.

The molar amounts of carbon and oxygen in a 100-g sample are calculated by dividing each element’s mass by its molar mass:

Coefficients for the tentative empirical formula are derived by dividing each molar amount by the lesser of the two:

Since the resulting ratio is one carbon to two oxygen atoms, the empirical formula is CO 2 .

Derivation of Molecular Formulas

Recall that empirical formulas are symbols representing the relative numbers of a compound’s elements. Determining the absolute numbers of atoms that compose a single molecule of a covalent compound requires knowledge of both its empirical formula and its molecular mass or molar mass. These quantities may be determined experimentally by various measurement techniques. Molecular mass, for example, is often derived from the mass spectrum of the compound (see discussion of this technique in the previous chapter on atoms and molecules). Molar mass can be measured by a number of experimental methods, many of which will be introduced in later chapters of this text.

Molecular formulas are derived by comparing the compound’s molecular or molar mass to its empirical formula mass . As the name suggests, an empirical formula mass is the sum of the average atomic masses of all the atoms represented in an empirical formula. If the molecular (or molar) mass of the substance is known, it may be divided by the empirical formula mass to yield the number of empirical formula units per molecule ( n ):

The molecular formula is then obtained by multiplying each subscript in the empirical formula by n , as shown by the generic empirical formula A x B y :

For example, consider a covalent compound whose empirical formula is determined to be CH 2 O. The empirical formula mass for this compound is approximately 30 amu (the sum of 12 amu for one C atom, 2 amu for two H atoms, and 16 amu for one O atom). If the compound’s molecular mass is determined to be 180 amu, this indicates that molecules of this compound contain six times the number of atoms represented in the empirical formula:

Molecules of this compound are then represented by molecular formulas whose subscripts are six times greater than those in the empirical formula:

Note that this same approach may be used when the molar mass (g/mol) instead of the molecular mass (amu) is used. In this case, one mole of empirical formula units and molecules is considered, as opposed to single units and molecules.

Example 3.13

Determination of the molecular formula for nicotine.

Next, calculate the molar ratios of these elements relative to the least abundant element, N.

The C-to-N and H-to-N molar ratios are adequately close to whole numbers, and so the empirical formula is C 5 H 7 N. The empirical formula mass for this compound is therefore 81.13 amu/formula unit, or 81.13 g/mol formula unit.

Calculate the molar mass for nicotine from the given mass and molar amount of compound:

Comparing the molar mass and empirical formula mass indicates that each nicotine molecule contains two formula units:

Finally, derive the molecular formula for nicotine from the empirical formula by multiplying each subscript by two:

C 8 H 10 N 4 O 2

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• Authors: Paul Flowers, Klaus Theopold, Richard Langley, William R. Robinson, PhD
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• Book title: Chemistry 2e
• Publication date: Feb 14, 2019
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• Book URL: https://openstax.org/books/chemistry-2e/pages/1-introduction
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Calculate Percent Error 5

Percent Error Definition

Percent error, sometimes referred to as percentage error, is an expression of the difference between a measured value and the known or accepted value . It is often used in science to report the difference between experimental values and expected values.

The formula for calculating percent error is:

Note: occasionally, it is useful to know if the error is positive or negative. If you need to know the positive or negative error, this is done by dropping the absolute value brackets in the formula. In most cases, absolute error is fine. For example, in experiments involving yields in chemical reactions, it is unlikely you will obtain more product than theoretically possible.

Steps to Calculate the Percent Error

• Subtract the accepted value from the experimental value.
• Take the absolute value of step 1
• Divide that answer by the accepted value.
• Multiply that answer by 100 and add the % symbol to express the answer as a percentage .

Example Calculation

Now let’s try an example problem.

You are given a cube of pure copper. You measure the sides of the cube to find the volume and weigh it to find its mass. When you calculate the density using your measurements, you get 8.78 grams/cm 3 . Copper’s accepted density is 8.96 g/cm 3 . What is your percent error?

Solution: experimental value = 8.78 g/cm 3 accepted value = 8.96 g/cm 3

Step 1: Subtract the accepted value from the experimental value.

8.78 g/cm 3 – 8.96 g/cm 3 = -0.18 g/cm 3

Step 2: Take the absolute value of step 1

|-0.18 g/cm 3 | = 0.18 g/cm 3

Step 3: Divide that answer by the accepted value.

Step 4: Multiply that answer by 100 and add the % symbol to express the answer as a percentage.

0.02 x 100 = 2 2%

The percent error of your density calculation is 2%.

Related Posts

5 thoughts on “ calculate percent error ”.

Percent error is always represented as a positive value. The difference between the actual and experimental value is always the absolute value of the difference. |Experimental-Actual|/Actualx100 so it doesn’t matter how you subtract. The result of the difference is positive and therefore the percent error is positive.

Percent error is always positive, but step one still contains the error initially flagged by Mark. The answer in that step should be negative:

experimental-accepted=error 8.78 – 8.96 = -0.18

In the article, the answer was edited to be correct (negative), but the values on the left are still not in the right order and don’t yield a negative answer as presented.

Mark is not correct. Percent error is always positive regardless of the values of the experimental and actual values. Please see my post to him.

Say if you wanted to find acceleration caused by gravity, the accepted value would be the acceleration caused by gravity on earth (9.81…), and the experimental value would be what you calculated gravity as :)

If you don’t have an accepted value, the way you express error depends on how you are making the measurement. If it’s a calculated value, like, based on a known about of carbon dioxide dissolved in water, then you have a theoretical value to use instead of the accepted value. If you are performing a chemical reaction to quantify the amount of carbonic acid, the accepted value is the theoretical value if the reaction goes to completion. If you are measuring the value using an instrument, you have uncertainty of the instrument (e.g., a pH meter that measures to the nearest 0.1 units). But, if you are taking measurements, most of the time, measure the concentration more than once, take the average value of your measurements, and use the average (mean) as your accepted value. Error gets complicated, since it also depends on instrument calibration and other factors.

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Chapter 3: Molecules, Compounds, and Chemical Equations

Back to chapter, experimental determination of chemical formula, previous video 3.8: formula mass and mole concepts of compounds, next video 3.10: chemical equations.

Chemical compounds are usually described using an empirical- or molecular formula. These formulas provide information about the amount of the different atoms of elements involved. But how are these formulas established?

Experimental analysis, like decomposition of compounds, is used to estimate the relative masses of constituent elements in the compound. These relative masses are then used to calculate the number of moles of each element to determine the formula of a chemical compound.

For example, a sample of a compound is experimentally determined to contain 43.64 grams of phosphorus and 56.36 grams of oxygen. Using molar masses as conversion factors, the relative masses from the experimental data are converted to 1.41 moles for phosphorus and 3.52 moles for oxygen. These mole values, when assigned as provisional subscripts to the elements, yield a pseudo-formula of the compound.

Dividing the mole values by the smallest mole value provides the mole ratios of approximately 2.5 moles oxygen to 1 mole phosphorus, which directly relates to the actual proportion of elements in the compound. 

If one of the quotients is still a decimal, then all the numbers are multiplied by the smallest counting number that gives the smallest whole-number ratio of subscripts, generating the empirical formula of P 2 O 5 . 

The molecular formula of compounds can be determined from their empirical formula and either the molar mass or molecular weight.

For example, the chemical compound with the empirical formula of P 2 O 5 is experimentally measured to have a molar mass of 283.89 g/mol. Its molecular formula is a whole-number multiple of its empirical formula, while its molar mass is a whole-number multiple of its empirical formula mass.

The ratio of molar mass and empirical formula mass yields the number of formula units. Multiplying empirical formula with the number of formula units gives the molecular formula. Hence, from the molecular formula P 4 O 10 , the compound is identified as tetraphosphorus decaoxide, or more commonly known by its empirical formula name as phosphorus pentoxide.

The elemental makeup of a compound defines its chemical identity, and chemical formulas are the most concise way of representing this elemental makeup. When a compound’s formula is unknown, measuring the mass of its constituent elements is often the first step in determining the formula experimentally.

Determination of Empirical Formulas

The most common approach to determining a compound’s chemical formula is first to measure the masses of its constituent elements. However, chemical formulas represent the relative numbers and not masses of atoms in the substance. Therefore, any experimentally derived data involving mass must be used to obtain the corresponding numbers of atoms in the compound. This is accomplished using molar masses to convert the mass of each element to its number of moles. These molar amounts are used to compute whole-number ratios that can be used to derive the empirical formula of the substance.

Consider a sample of a compound determined to contain 1.71 grams of carbon and 0.287 grams of hydrogen. The corresponding numbers of atoms are 0.142 moles of carbon and 0.284 moles of hydrogen. Thus, this compound may be represented by the formula C 0.142 H 0.284 . Per convention, formulas contain whole-number subscripts, which can be achieved by dividing each subscript by the smallest subscript (0.142). The empirical formula for this compound is thus CH 2 . Subscripts of “1” are not written but rather assumed if no other number is present. This may or not be the compound’s molecular formula; however, additional information is needed to make that determination.

As a second example, a sample of a compound is determined to contain 5.31 grams of chlorine and 8.40 grams of oxygen. The same approach yields a tentative empirical formula of ClO 3.5 . In this case, dividing by the smallest subscript still leaves a decimal in the empirical formula. To convert this into a whole number, multiply each of the subscripts by two, retaining the same atom ratio and yielding Cl 2 O 7 as the final empirical formula.

Deriving Empirical Formulas from Percent Composition

In instances where the percent composition of a compound is available, it is used to calculate the masses of elements present in the compound. Since the scale for percentages is 100, it is convenient to calculate the mass of elements present in a sample weighing 100 grams. The masses obtained are used to derive the empirical formula.

For example, suppose a gaseous compound contains 27.29% C and 72.71% O. The mass percentages, therefore, are expressed as fractions:

The mass of carbon, 27.29 g, corresponds to 2.272 moles of carbon, and the mass of oxygen, 72.71 g, corresponds to 4.544 moles of oxygen. The representative formula is, therefore, C 2.272 O 4.544 . Dividing each subscript with 2.272 provides the empirical formula: CO 2 . 

Derivation of Molecular Formulas

Determining the absolute numbers of atoms that compose a single molecule of a covalent compound requires knowledge of both its empirical formula and its molecular mass or molar mass. These quantities may be determined experimentally by various measurement techniques. Molecular mass, for example, is often derived from the mass spectrum of the compound.

Molecular formulas are derived by comparing the compound’s molar mass or molecular mass to its empirical formula mass. As the name suggests, an empirical formula mass is the sum of the average atomic masses of all the atoms represented in an empirical formula. If the known molar mass of a substance is divided by the empirical formula mass, it yields the number of empirical formula units per molecule ( n ). 

The molecular formula is then obtained by multiplying each subscript in the empirical formula by n , as shown by the generic empirical formula A x B y :

For example, the empirical formula of a covalent compound is determined to be CH 2 O, and its empirical formula mass is approximately 30 amu. If the compound’s molecular mass is determined to be 180 amu, this indicates that molecules of this compound contain six times the number of atoms represented in the empirical formula. 

Molecules of this compound are then represented by a molecular formula with subscripts that are six times greater than those in the empirical formula: (CH 2 O) 6 = C 6 H 12 O 6 .

This text is adapted from Openstax, Chemistry 2e, Section 3.2: Determining Empirical and Molecular Formulas.

How to Calculate Percent Error

What Is the Formula for Percent Error?

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Percent error or percentage error expresses the difference between an approximate or measured value and an exact or known value as a percentage. It is a well-known type of error calculation, along with absolute and relative error.

Percent error plays a crucial role in validating hypotheses and assessing the accuracy of measurements in scientific research, and it also plays a fundamental role in quality control processes, where deviations from expected values could signify potential flaws in manufacturing or experimental procedures.

Here is the formula used to calculate percent error, along with an example calculation.

Key Points: Percent Error

• The purpose of a percent error calculation is to gauge how close a measured value is to a true value.
• Percent error is equal to the difference between an experimental and theoretical value, divided by the theoretical value, and then multiplied by 100 to give a percent.
• In some fields, percent error is always expressed as a positive number. In others, it is correct to have either a positive or negative value. The sign helps determine whether recorded values consistently fall above or below expected values.

Percent Error Formula

Percent error is the difference between a measured or experiment value and an accepted or known value, divided by the known value, multiplied by 100%.

For many applications, percent error is always expressed as a positive value. The absolute value of the error is divided by an accepted value and given as a percent.

Percent Error = | Accepted Value - Experimental Value | / Accepted Value x 100%

For chemistry and other sciences, it is customary to keep a negative value, should one occur. Whether error is positive or negative is important. For example, you would not expect to have a positive percent error comparing actual to theoretical yield in a chemical reaction . If a positive value was calculated, this would give clues as to potential problems with the procedure or unaccounted reactions.

When keeping the sign for error, the calculation is the experimental or measured value minus the known or theoretical value, divided by the theoretical value and multiplied by 100%.

Percent Error = [Experimental Value - Theoretical Value] / Theoretical Value x 100%

Percent Error Calculation Steps

• Subtract one value from another. The order does not matter if you are dropping the sign (taking the absolute value. Subtract the theoretical value from the experimental value if you are keeping negative signs. This value is your "error."
• Divide the error by the exact or ideal value (not your experimental or measured value). This will yield a decimal number.
• Convert the decimal number into a percentage by multiplying it by 100.
• Add a percent or % symbol to report your percent error value.

Percent Error Example Calculation

In a lab, you are given a block of aluminum . You measure the dimensions of the block and its displacement in a container of a known volume of water. You calculate the density of the block of aluminum to be 2.68 g/cm 3 . You look up the density of a block of aluminum at room temperature and find it to be 2.70 g/cm 3 . Calculate the percent error of your measurement.

• Subtract one value from the other: 2.68 - 2.70 = -0.02
• Depending on what you need, you may discard any negative sign (take the absolute value): 0.02 This is the error.
• Divide the error by the true value: 0.02/2.70 = 0.0074074
• Multiply this value by 100% to obtain the percent error: 0.0074074 x 100% = 0.74% (expressed using two significant figures ). Significant figures are important in science. If you report an answer using too many or too few, it may be considered incorrect, even if you set up the problem properly.

Percent Error vs. Absolute and Relative Error

Percent error is related to absolute error and relative error . The difference between an experimental and known value is the absolute error. When you divide that number by the known value you get relative error . Percent error is relative error multiplied by 100%. In all cases, report values using the appropriate number of significant digits.

Why Is Percent Error Important?

Percent error is used extensively across various fields such as physics, chemistry, engineering, and statistics. Because it measures deviations from a true value or accepted value, percent error can be utilized to validate hypotheses during experiments or ensure quality control in manufacturing processes.

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Percent Error / Percent Difference: Definition, Examples

Statistics Definitions >

• Percent Error
• Percent Difference

What is Percent Error?

Percent errors tells you how big your errors are when you measure something in an experiment. Smaller values mean that you are close to the accepted or real value. For example, a 1% error means that you got very close to the accepted value, while 45% means that you were quite a long way off from the true value. Measurement errors are mostly unavoidable: equipment can be imprecise, hands can shake, or your instruments just might not have the capability to measure accurately. Percent error will let you know how badly these unavoidable errors affected your results.

The formula is:

PE = (|accepted value – experimental value| \ accepted value) x 100%. Example question: The accepted distance to the moon is 238,855 miles.* You measure the distance as 249,200 miles. What is the percent error? Solution: Step 1: Insert your data into the formula: PE = (|accepted value – experimental value| \ accepted value) x 100% = ((|238,855 miles – 249,200|) \ 238,855 miles) x 100% = Step 2: Solve: (10345 \ 238,855 miles) x 100% = 0.0433 * 100% = 4.33%.

*That’s the average distance, but let’s assume it’s the distance on the day you’re taking the measurement!

Note : in some sciences, the absolute value sign is sometimes (but not always) omitted. You may want to refer to your textbook to see if the author is omitting the absolute value sign. If you aren’t sure, the most common form is with the absolute value sign.

Alternate Wording

Accepted value is sometimes called the “true” value or “theoretical” value, so you might see the formula written in slightly different ways:

• PE = (|true value – experimental value| \ true value) x 100%.
• PE = (|theoretical value – experimental value| \ theoretical value) x 100%.

All three versions of the formula mean the exact same thing — it’s just different wording.

Alternative Definition of Percent Error using Relative Error

The percentage error is sometimes reported as being 100% times the relative error . Be careful though, because there are actually two types of relative error : one for precision and one for accuracy (not sure of the difference between the two? See: Accuracy and Precision ). The definition “100% times the relative error” is only true if you are using the “accuracy” version of relative error:

• RE accuracy = (Absolute error / “True” value) * 100%.

The definition does not work if you’re using the RE for precision:

• RE precision = absolute error / measurement being taken.

What is Percent Difference?

• E 1 is the first experimental measurement.
• E 2 is the second experimental measurement.

Example question: You make two measurements in an experiment of 21 mL and 22 mL. What is the percent difference?

Experimental Probability

Grade 7 math worksheets.

Experimental probability refers to the probability of an event based on actual experimentation or observation of outcomes.

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Experimental probability refers to the probability of an event based on actual experimentation or observation of outcomes. It is determined by conducting an experiment or observing an event multiple times and recording the number of times the event occurs.

To find the experimental probability of an event, you would divide the number of times the event occurred by the total number of trials or observations. For example, if you flipped a coin 20 times and it landed on heads 12 times, the experimental probability of flipping heads would be 12/20 or 0.6.

Experimental probability is often used in situations where it is difficult or impossible to determine the theoretical probability of an event. It can be used to estimate the theoretical probability, but it may not be as accurate as using mathematical formulas to calculate probability.

However, experimental probability can still provide valuable information about the likelihood of an event occurring, especially if the sample size is large enough to reduce the effects of randomness and variability.

Experimental Probability Examples:

Example 1: You roll a six-sided die 100 times and record the number of times each number comes up. You find that the number 3 comes up 23 times. The experimental probability of rolling a 3 on the die is therefore 23/100 or 0.23.  

Example 2: You toss a coin 50 times and record the number of times it lands on heads. You find that it lands on heads 27 times. The experimental probability of flipping heads is therefore 27/50 or 0.54.

Example 3: You draw a card from a deck of 52 cards 200 times and record the number of times you draw a heart. You find that you draw a heart 45 times. The experimental probability of drawing a heart is therefore 45/200 or 0.225.

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Experimental Probability Formula

Here are some common formulas used to calculate probability:

Theoretical Probability Formula: Theoretical probability is the probability of an event based on mathematical calculations. The formula for theoretical probability is:

P(A) = Number of favorable outcomes / Total number of outcomes

where P(A) represents the probability of event A.

Experimental Probability Formula: Experimental probability is the probability of an event based on actual experimentation or observation. The formula for experimental probability is:

P(A) = Number of times event A occurs / Total number of trials or observations

Conditional Probability Formula: Conditional probability is the probability of an event occurring given that another event has already occurred. The formula for conditional probability is:

P(A | B) = P(A and B) / P(B)

where P(A | B) represents the probability of event A given that event B has occurred, P(A and B) represents the probability of both A and B occurring, and P(B) represents the probability of event B occurring.

Multiplication Rule Formula: The multiplication rule is used to calculate the probability of two or more independent events occurring together. The formula for the multiplication rule is:

P(A and B) = P(A) * P(B)

where P(A and B) represents the probability of both A and B occurring, and P(A) and P(B) represent the probabilities of events A and B occurring, respectively.

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Experimental Probability FAQS

What is probability.

Probability is a measure of the likelihood of an event occurring. It is represented as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

What are the types of probability?

There are two main types of probability: theoretical probability and experimental probability. Theoretical probability is based on mathematical calculations, while experimental probability is based on actual experimentation or observation.

What is conditional probability?

Conditional probability is the probability of an event occurring given that another event has already occurred. It is calculated using the formula P(A | B) = P(A and B) / P(B), where P(A | B) represents the probability of event A given that event B has occurred.

What is the difference between independent and dependent events?

Independent events are events in which the occurrence of one event does not affect the probability of the other event occurring. Dependent events are events in which the occurrence of one event affects the probability of the other event occurring.

What is the law of large numbers?

The law of large numbers states that as the number of trials or observations increases, the experimental probability of an event approaches its theoretical probability. This means that with a large enough sample size, the experimental probability becomes more accurate.

Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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Science in School

Classic chemistry: finding the empirical formula teach article.

Author(s): Caroline Evans

Witness a spectacular chemical reaction and take some careful measurements to work out the empirical formula of a compound.

Every chemical compound has a chemical formula. In fact, there are several different types of chemical formula for any one compound (figure 1). Perhaps the most familiar type is the molecular formula – such as H 2 O for water and CO 2 for carbon dioxide – which tells us the number of different atoms in each molecule. Structural formulae go a step further by showing how the atoms are linked together within the molecule, which is especially important for organic compounds.

The simplest type of formula – called the empirical formula – shows just the ratio of different atoms. For example, while the molecular formula for glucose is C 6 H 12 O 6 , its empirical formula is CH 2 O – showing that there are twice as many hydrogen atoms as carbon or oxygen atoms, but not the actual numbers of atoms in a single molecule or how they are arranged. These simple, ratio-based formulae were developed by early chemists in the 18th century. They are known as ‘empirical’ formulae because the ratio between the numbers of atoms in a compound can be found by traditional methods of chemical analysis by experiment.

Today, working out an empirical formula experimentally is an important feature of chemistry courses all over the world. It is also the first step in working out the chemical formula of an unidentified compound, making it a useful tool in chemical analysis. The classic school demonstration involves heating magnesium in a crucible to make magnesium oxide – a dull white powder. In this article, a much more exciting alternative is described: a dramatic reaction between tin and iodine, producing a bold purple vapour and bright orange crystals as the reaction progresses.

The aim of the experiment is to calculate the ratio between tin and iodine atoms in tin iodide. This is done by synthesising this compound and accurately measuring the mass of the reagents at the start of the experiment and the leftover tin at the end. The experiment involves a range of techniques, including setting up and using a reflux condenser and using organic solvents for extraction. As well as covering the practical exercise of deriving the empirical formula, the experiment links to more theoretical chemistry topics, such as the benefits of using a reagent in excess, the synthesis of compounds from their elements, and how bonding can be linked to solubility. It’s also a very clear application of the law of conservation of mass, which is a fundamental principle throughout chemistry (and science generally).

Depending on the number of fume cupboards available in your classroom, students can carry out the experiment themselves, but it is also suitable as a teacher demonstration. The experiment takes about two hours and works best in a double lesson, but it is also possible to split it between two single lessons. It is most suitable for students aged 16–18, but it could also be used as an extension activity for those aged 14–16.

The teacher (or each group of students) will need the following:

• 3 g iodine
• 5 g tin metal, in pieces no larger than about 1 cm square
• 60 ml cyclohexane
• 250 ml round-bottomed flask (for example Quickfit®)
• 100 ml measuring cylinder
• Two 250 ml beakers
• Electric heating mantle
• Reflux condenser
• Water supply
• 2 large pieces of filter paper
• Accurate weighing balance
• Lab jack (optional)

Safety note

Students should wear a lab coat, gloves and safety goggles. Solid iodine is corrosive and can stain the skin, which is why gloves should be worn. The experiment should be carried out in a fume cupboard. As iodine vapour is toxic, ensure that the purple vapour does not rise more than one-third of the way up the reflux condenser when heating. Cyclohexane and propanone are highly flammable, so a heating mantle is required, and care should also be taken to keep both these solvents away from naked flames. Propanone should be disposed of in a solvent residues bottle. In addition, teachers should follow their local health and safety rules.

• Place the 250 ml round-bottomed flask on weighing scales. Place about 3 g of solid iodine in the flask, and record accurately the mass added.
• Add about 5 g of tin metal to the flask and record accurately the mass of tin used.

• Lower the flask into an electric heating mantle. You may wish to use a lab jack to allow you to raise and lower the heater when required.
• Attach a reflux condenser vertically into the neck of the flask, then clamp and connect this to the water supply.
• Heat the mixture gently in the flask until it starts to boil.
• Now reduce the heat so that the mixture boils steadily and the purple iodine vapour rises no more than one-third of the way up the condenser.
• Continue heating until there is no longer any trace of purple, and the liquid dripping back into the flask from the condenser is colourless. The liquid in the flask should be orange. This may take up to an hour to complete.

• In a fume cupboard, pour the orange liquid from the flask into a beaker, taking care not to tip out any of the residual tin metal.
• Pour a small amount of propanone into the round-bottomed flask and swirl this around to dissolve any remaining tin iodide that could still be in the flask with the excess solid tin. Carefully decant the propanone washings into another beaker. Repeat this process of swirling with propanone until the propanone poured from the flask into the beaker is colourless. This step ensures the leftover solid tin does not contain any of the orange tin iodide.
• Leave the beaker with the orange liquid (step 10) in the fume cupboard overnight so that the solvent evaporates, allowing beautiful orange tin iodide crystals to form. (You can look at these in the next lesson.)

Determining the empirical formula

This experiment involves reacting two substances – tin and iodine – in their elemental form to produce the compound tin iodide. Tin has more than one possible oxidation state, so the reaction could produce either tin(II) iodide (SnI 2 ) or tin(IV) iodide (SnI 4 ). Using the experimental data, we can derive the empirical formula for the product, which will tell us the ratio between tin and iodine. From this, we can work out the identity of the tin compound produced.

• The first step is to work out the masses of the iodine and tin used in the reaction. The iodine is all used in the reaction (as it is the limiting reagent, while tin is in excess), so we can use the accurate mass of iodine weighed in step 1.
• To calculate the mass of tin used in the reaction, we need to subtract the mass of tin left over once the reaction has finished from the initial mass.

Tin used in the reaction = initial mass (step 2) minus the leftover mass (step 13)

• Now we need to convert masses into moles, to find the amount of tin and iodine atoms used in the reaction. To do this, we divide each mass value (from steps 14 and 15) by the relative atomic mass (A r ) of the element. The resulting answers will tell us the number of moles of each element used in the reaction (and thus in the final compound).
• To find the empirical formula of tin iodide, we need to find the number of moles of iodine used for one mole of tin. Here, we divide both answers in step 16 by the number of moles of tin used (as this will tell us how many moles of iodine combine with one mole of tin). You may need to round this ratio slightly to form simple whole numbers: for example, a ratio of tin to iodine of 1:3.6 can be rounded to 1:4.
• Finally, you can write the empirical formula: for example, a ratio of 1:4 means that the empirical formula is SnI 4 . What empirical formula did you find?

This activity can produce good results if carried out carefully, with values that should quite closely round to 1:4 (as the ratio of tin to iodine). This leads to SnI 4 as the empirical formula for tin iodide.

In general, values ranging between 1:3.2 and 1:3.8 are often obtained. As the common oxidation states for tin are +2 and +4, an experimental outcome giving a ratio close to 1:3 would not be in agreement. However, such values can open up a discussion about sources of experimental error and the importance of precision.

Sources of error in experimental measurements

After the experiment, ask all students to think about possible sources of error. What effect might each of the following have on the final results?

• Accuracy of the balance used
• Incomplete reaction of iodine
• Loss of iodine vapour from the condenser
• Poor washing of the residual tin
• Loss of tin while washing with propanone
• Incomplete drying of the tin before recording the mass

Table 1 summarises the effect of each of these sources of error in the experiment and on the final result – that is, how each changes the value of x in the empirical formula SnI x .

Extension discussion: solubility and bonding

This experiment also offers an opportunity to discuss how bonding is linked to solubility. Iodine and tin iodide both dissolve in non-polar solvents (cyclohexane and propanone) but not in water, whereas tin is a metal and is insoluble in cyclohexane, propanone and water. Using this information, can your students draw conclusions about the likely bonding in these substances?

Acknowledgement

The author would like to thank Alan Carter, who was the Head of Chemistry at Wellington College (Berkshire, UK) until 2004, and who created the initial resource that inspired this article.

• Find out more about the oxidation states of tin at the Chemguide website .
• Learn about an alternative experiment for deducing an empirical formula, this time for copper(II) oxide, at the Royal Society of Chemistry website .

Caroline Evans is the Head of Chemistry at Wellington College, Berkshire, UK. She is a reviewer for Science in School and has reviewed numerous textbooks linked to A-level and International Baccalaureate courses. Caroline is also an examiner for both GCSE and A-level chemistry.

This article describes a spectacular experiment to work out the empirical formula of a compound produced from its elements. This practical exercise gives students an opportunity to go beyond numerical exercises when working out chemical formulae.

The experiment is suitable for senior chemistry students studying analytical chemistry. It involves a wide range of experimental techniques and could be used as a starting point for discussing different sources of error in experimental measurements. It can also provide a basis for other key topics, including how bonding is linked to solubility.

All the materials required can be easily obtained and the instructions are easy to follow, making the activities suitable for students to perform in groups.

Mireia Güell Serra, chemistry and mathematics teacher, INS Cassà de la Selva school, Spain

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• Math Article
• Experimental Probability

You and your 3 friends are playing a board game. It’s your turn to roll the die and to win the game you need a 5 on the dice. Now, is it possible that upon rolling the die you will get an exact 5? No, it is a matter of chance. We face multiple situations in real life where we have to take a chance or risk. Based on certain conditions, the chance of occurrence of a certain event can be easily predicted. In our day to day life, we are more familiar with the word ‘ chance and probability ’. In simple words, the chance of occurrence of a particular event is what we study in probability. In this article, we are going to discuss one of the types of probability called  “Experimental Probability” in detail.

What is Probability?

Probability, a branch of Math that deals with the likelihood of the occurrences of the given event. The probability values for the given experiment is usually defined between the range of numbers. The values lie between the numbers 0 and 1. The probability value cannot be a negative value. The basic rules such as addition, multiplication and complement rules are associated with the probability.

Experimental Probability Vs Theoretical Probability

There are two approaches to study probability:

• Theoretical Probability

What is Experimental Probability?

Experimental probability, also known as Empirical probability, is based on actual experiments and adequate recordings of the happening of events. To determine the occurrence of any event, a series of actual experiments are conducted. Experiments which do not have a fixed result are known as random experiments. The outcome of such experiments is uncertain. Random experiments are repeated multiple times to determine their likelihood. An experiment is repeated a fixed number of times and each repetition is known as a trial. Mathematically, the formula for the experimental probability is defined by;

Probability of an Event P(E) = Number of times an event occurs / Total number of trials.

What is Theoretical Probability?

In probability, the theoretical probability is used to find the probability of an event. Theoretical probability does not require any experiments to conduct. Instead of that, we should know about the situation to find the probability of an event occurring. Mathematically, the theoretical probability is described as the number of favourable outcomes divided by the number of possible outcomes.

Probability of Event P(E) = No. of. Favourable outcomes/ No. of. Possible outcomes.

Experimental Probability Example

Example: You asked your 3 friends Shakshi, Shreya and Ravi to toss a fair coin 15 times each in a row and the outcome of this experiment is given as below:

Calculate the probability of occurrence of heads and tails.

Solution: The experimental probability for the occurrence of heads and tails in this experiment can be calculated as:

Experimental Probability of Occurrence of heads = Number of times head occurs/Number of times coin is tossed.

Experimental Probability of Occurrence of tails = Number of times tails occurs/Number of times coin is tossed.

We observe that if the number of tosses of the coin increases then the probability of occurrence of heads or tails also approaches to 0.5.

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Experimental Probability

Experimental probability: introduction, experimental probability: definition, experimental probability formula, solved examples, practice problems, frequently asked questions.

In mathematics, probability refers to the chance of occurrence of a specific event. Probability can be measured on a scale from 0 to 1. The probability is 0 for an impossible event. The probability is 1 if the occurrence of the event is certain.

There are two approaches to study probability: experimental and theoretical. 

Suppose you and your friend toss a coin to decide who gets the first turn to ride a new bicycle. You choose “heads” and your friend chooses “tails.” 

Can you guess who will win? No! You have $\frac{1}{2}$ a chance of winning and so does your friend. This is theoretical since you are predicting the outcome based on what is expected to happen and not on the basis of outcomes of an experiment.

So, what is the experimental probability? Experimental probability is calculated by repeating an experiment and observing the outcomes. Let’s understand this a little better.

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Experimental probability, or empirical probability, is the probability calculated by performing actual experiments and gathering or recording the necessary information. How would you define an experiment? The math definition of an experiment is “a process or procedure that can be repeated and that has a set of well-defined possible results or outcomes.”

Consider the same example. Suppose you flip the coin 50 times to see whether you get heads or tails, and you record the outcomes. Suppose you get heads 20 times and tails 30 times. Then the probability calculated using these outcomes is experimental probability. Here, t he experimental meaning is connected with such experiments used to determine the probability of an event.

Now that you know the meaning of experimental probability, let’s understand its formula.

Experimental Probability for an Event A can be calculated as follows:

P(E) $= \frac{Number of occurance of the event A}{Total number of trials}$

Let’s understand this with the help of the last example. 

A coin is flipped a total of 50 times. Heads appeared 20 times. Now, what is the experimental probability of getting heads?

E xperimental probability of getting heads $= \frac{Number of occurrences}{Total number of trials}$

P (Heads) $= \frac{20}{50} = \frac{2}{5}$

P (Tails) $= \frac{30}{50} = \frac{3}{5}$

Experimental Probability vs. Theoretical Probability

Theoretical probability expresses what is expected. On the other hand, experimental probability explains how frequently an event occurred in an experiment.

If you roll a die, the theoretical probability of getting any particular number, say 3, is $\frac{1}{6}$. 

However, if you roll the die 100 times and record how many times 3 appears on top, say 65 times, then the experimental probability of getting 3 is $\frac{65}{100}$.

Theoretical probability for Event A can be calculated as follows:

P(A) $= \frac{Number of outcomes favorable to Event A}{Number of possible outcomes}$

In the example of flipping a coin, the theoretical probability of the occurrence of heads (or tails) on tossing a coin is

P(H) $= \frac{1}{2}$ and  P(T) $= \frac{1}{2}$ (since possible outcomes are $2 -$ head or tail)

Experimental Probability: Examples

Let’s take a look at some of the examples of experimental probability .

Example 1: Ben tried to toss a ping-pong ball in a cup using 10 trials, out of which he succeeded 4 times. 

P(win) $= \frac{Number of success}{Number of trials}$

             $= \frac{4}{10}$

             $= \frac{2}{5}$

Example 2: Two students are playing a game of die. They want to know how many times they land on 2 on the dice if the die is rolled 20 times in a row. 

The experimental probability of rolling a 2 

$= \frac{Number of times 2 appeared}{Number of trials}$

$= \frac{5}{20}$

$= \frac{1}{4}$

1. Probability of an event always lies between 0 and 1.

2. You can also express the probability as a decimal and a percentage.

Experimental probability is a probability that is determined by the results of a series of experiments. Learn more such interesting concepts at SplashLearn .

1. Leo tosses a coin 25 times and observes that the “head” appears 10 times. What is the experimental probability of getting a head?

 P(Head) $= \frac{Number of times heads appeared}{Total number of trials}$

               $= \frac{10}{25}$

               $= \frac{2}{5}$

               $= 0.4$

2. The number of cakes a baker makes per day in a week is given as 7, 8, 6, 10, 2, 8, 3. What is the probability that the baker makes less than 6 cakes the next day?

Solution: 

Number of cakes baked each day in a week $= 7, 8, 6, 10, 2, 8, 3$

Out of 7 days, there were 2 days (highlighted in bold) on which the baker made less than 6 cookies.

P$(< 6 $cookies$) = \frac{2}{7}$

3. The chart below shows the number of times a number was shown on the face of a tossed die. What was the probability of getting a 3 in this experiment?

Number of times 3 showed $= 7$

Number of tosses $= 30$

P(3) $= \frac{7}{30}$

4. John kicked a ball 20 times. He kicked 16 field goals and missed 4 times . What is the experimental probability that John will kick a field goal during the game?

Solution:  

John succeeded in kicking 16 field goals. He attempted to kick a field goal 20 times. 

So, the number of trials $= 20$

John’s experimental probability of kicking a field goal $= \frac{Successful outcomes} {Trials attempted} = \frac{16}{20}$ 

$= \frac{4}{5}$

$= 0.8$ or $80%$

5. James recorded the color of bikes crossing his street. Of the 500 bikes, 10 were custom colors, 100 were white, 50 were red, 120 were black, 100 were silver, 60 were blue, and 60 were gray. What is the probability that the car crossing his street is white?

Number of white bikes $= 100$ 

Total number of bikes $= 500$

P(white bike) $=  \frac{100}{500} = \frac{1}{5}$

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In a class, a student is chosen randomly in five trials to participate in 5 different events. Out of chosen students, 3 were girls and 2 were boys. What is the experimental probability of choosing a boy in the next event?

A manufacturer makes 1000 tablets every month. after inspecting 100 tablets, the manufacturer found that 30 tablets were defective. what is the probability that you will buy a defective tablet, the 3 coins are tossed 1000 times simultaneously and we get three tails $= 160$, two tails $= 260$, one tail $= 320$, no tails $= 260$. what is the probability of occurrence of two tails, the table below shows the colors of shirts sold in a clothing store on a particular day and their respective frequencies. use the table to answer the questions that follow. what is the probability of selling a blue shirt.

Jason leaves for work at the same time each day. Over a period of 327 working days, on his way to work, he had to wait for a train at the railway crossing for 68 days. What is the experimental probability that Jason has to wait for a train on his way to work?

What is the importance of experimental probability?

Experimental probability is widely used in research and experiments in various fields, such as medicine, social sciences, investing, and weather forecasting.

Is experimental probability always accurate?

Predictions based on experimental probability are less reliable than those based on theoretical probability.

Can experimental probability change every time the experiment is performed?

Since the experimental probability is based on the actual results of an experiment, it can change when the results of an experiment change.

What is theoretical probability?

The theoretical probability is calculated by finding the ratio of the number of favorable outcomes to the total number of probable outcomes.

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• Experimental Probability

Experimental probability , also known as empirical probability , is a concept in mathematics that deals with estimating the likelihood of an event occurring based on actual experimental results. Unlike theoretical probability , which predicts outcomes based on known possibilities, experimental probability is derived from real-life experiments and observations.

To understand this better, imagine flipping a coin. The theoretical probability of landing heads is 50% or 1/2. However, if you actually flip the coin 100 times and record the outcomes, you might get heads 48 times. The experimental probability of getting heads would then be 48/100 or 0.48.

In this article, we will explore the concept of experimental probability, its significance, and how it differs from theoretical probability. We will discuss the formula for calculating experimental probability, provide examples to illustrate its application.

Table of Content

What is Probability?

What is experimental probability, formula for experimental probability, examples of experimental probability, what is theoretical probability, experimental probability vs theoretical probability.

• Solved Examples
• Practice Problems

The branch of mathematics that tells us about the likelihood of the occurrence of any event is the probability . Probability tells us about the chances of happening an event.

The probability of any element that is sure to occur is One(1) whereas the probability of any impossible event is Zero(0). The probability of all the elements ranges between 0 to 1.

There are two ways of studying probability that are

• Theoretical Probability

Now let’s learn about both in detail.

Experimental probability is a type of probability that is calculated by conducting an actual experiment or by performing a series of trials to observe the occurrence of an event. It is also known as empirical probability.

To calculate experimental probability, you need to conduct an experiment by repeating the event multiple times and observing the outcomes. Then, you can find the probability of the event occurring by dividing the number of times the event occurred by the total number of trials.

The experimental Probability for Event A can be calculated as follows:

P(E) = (Number of times an event occur in an experiment) / (Total number of Trials)

Now, as we learn the formula, let’s put this formula in our coin-tossing case.  If we tossed a coin 10 times and recorded a head 4 times and a tail 6 times then the Probability of Occurrence of Head on tossing a coin:

P(H) = 4/10

Similarly, the Probability of Occurrence of Tails on tossing a coin:

P(T) = 6/10

Theoretical Probability deals with assumptions in order to avoid unfeasible or expensive repetition experiments. The theoretical Probability for an Event A can be calculated as follows:

P(A) = Number of outcomes favorable to Event A / Number of all possible outcomes            

Now, as we learn the formula, let’s put this formula in our coin-tossing case. In tossing a coin, there are two outcomes: Head or Tail.

Hence, The Probability of occurrence of Head on tossing a coin is

Similarly, The Probability of the occurrence of a Tail on tossing a coin is

There are some key differences between Experimental and Theoretical Probability , some of which are as follows:

• Probability in Maths
• Probability Distribution
• Bayes’ Theorem

Solved Examples of Experimental Probability

Example 1. Let’s take an example of tossing a coin, tossing it 40 times , and recording the observations. By using the formula, we can find the experimental probability for heads and tails as shown in the below table.

Number of Trail Outcome Number of Trail Outcome Number of Trail Outcome Number of Trail Outcome First H Eleventh T Twenty-first T Thirty-first T Second T Twelfth T Twenty-second H Thirty-second H Third T Thirteenth H Twenty-third T Thirty-third T Fourth H Fourteenth H Twenty-fourth H Thirty-fourth H Fifth H Fifteenth H Twenty-fifth T Thirty-fifth T Sixth H Sixteenth H Twenty-sixth H Thirty-sixth T Seventh T Seventeenth T Twenty-seventh T Thirty-seventh T Eighth H Eighteenth T Twenty-eighth T Thirty-eighth H Ninth T Nineteenth T Twenty-ninth T Thirty-ninth T Tenth H Twentieth T Thirtieth H Fortieth T The formula for experimental probability: P(H) = Number of Heads ÷ Total Number of Trials = 16 ÷ 40 = 0.4 Similarly, P(H) = Number of Tails ÷ Total Number of Trials = 24 ÷ 40 = 0.6 P(H) + P(T) = 0.6 + 0.4 = 1 Note: Repeat this experiment for ‘n’ times and then you will find that the number of times increases, the fraction of experimental probability comes closer to 0.5. Thus if we add P(H) and P(T), we will get  0.6 + 0.4 = 1 which means P(H) and P(T) is the only possible outcomes.

Example 2. A manufacturer makes 50,000 cell phones every month. After inspecting 1000 phones, the manufacturer found that 30 phones are defective. What is the probability that you will buy a phone that is defective? Predict how many phones will be defective next month.

Experimental Probability = 30/1000 = 0.03 0.03 = (3/100) × 100 = 3% The probability that you will buy a defective phone is 3% ⇒ Number of defective phones next month = 3% × 50000 ⇒ Number of defective phones next month = 0.03 × 50000 ⇒ Number of defective phones next month = 1500

Example 3. There are about 320 million people living in the USA. Pretend that a survey of 1 million people revealed that 300,000 people think that all cars should be electric. What is the probability that someone chosen randomly does not like the electric car? How many people like electric cars?

Now the number of people who do not like electric cars is 1000000 – 300000 = 700000 Experimental Probability =  700000/1000000 = 0.7 And, 0.7 = (7/10) × 100 = 70% The probability that someone chose randomly does not like the electric car is 70% The probability that someone like electric cars is  300000/1000000 = 0.3 Let x be the number of people who love electric cars ⇒ x = 0.3 × 320 million ⇒ x = 96 million The number of people who love electric cars is 96 million.

Practice Problems on Experimental Probability

Problem 1: A coin is flipped 200 times, and it lands on heads 120 times. What is the experimental probability of getting heads?

Problem 2: A die is rolled 50 times, and the number 3 appears 8 times. What is the experimental probability of rolling a 3?

Problem 3: In a class survey, 150 students were asked if they prefer reading books or watching movies. 90 students said they prefer watching movies. What is the experimental probability that a randomly chosen student prefers watching movies?

Problem 4: A bag contains 5 red, 7 blue, and 8 green marbles. If 40 marbles are drawn at random with replacement, and 12 of them are red, what is the experimental probability of drawing a red marble?

Problem 5: A basketball player made 45 successful free throws out of 60 attempts. What is the experimental probability that the player will make a free throw?

Problem 6: During a game, a spinner is spun 80 times, landing on a specific section 20 times. What is the experimental probability of the spinner landing on that section?

FAQs on Experimental Probability

Define experimental probability..

Probability of an event based on an actual trail in physical world is called experimental probability.

How is Experimental Probability calculated?

Experimental Probability is calculated using the following formula:  P(E) = (Number of trials taken in which event A happened) / Total number of trials

Can Experimental Probability be used to predict future outcomes?

No,  experimental probability can’t be used to predict future outcomes as it only achives the theorectical value when the trails becomes infinity.

How is Experimental Probability different from Theoretical Probability?

 Theoretical probability is the probability of an event based on mathematical calculations and assumptions, whereas experimental probability is based on actual experiments or trials.

What are some Limitations of Experimental Probability?

There are some limitation of experimental probability, which are as follows: Experimental probability can be influenced by various factors, such as the sample size, the selection process, and the conditions of the experiment.  The number of trials conducted may not be sufficient to establish a reliable pattern, and the results may be subject to random variation.  Experimental probability is also limited to the specific conditions of the experiment and may not be applicable in other contexts.

Can Experimental Probability of an event be a negative number if not why?

As experimental probability is given by: P(E) = Number of trials taken in which event A happened/Total number of trials Thus, it can’t be negative as both number are count of something and counting numbers are 1, 2, 3, 4, …. and they are never negative.

What are Types of Probability?

There are two forms of calculating the probability of an event that are, Theoretical Probability Experimental Probability

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Experimental vs Observational Studies: Differences & Examples

Understanding the differences between experimental vs observational studies is crucial for interpreting findings and drawing valid conclusions. Both methodologies are used extensively in various fields, including medicine, social sciences, and environmental studies. 

Researchers often use observational and experimental studies to gather comprehensive data and draw robust conclusions about their investigating phenomena. 

This blog post will explore what makes these two types of studies unique, their fundamental differences, and examples to illustrate their applications.

What is an Experimental Study?

An experimental study is a research design in which the investigator actively manipulates one or more variables to observe their effect on another variable. This type of study often takes place in a controlled environment, which allows researchers to establish cause-and-effect relationships.

Key Characteristics of Experimental Studies:

• Manipulation: Researchers manipulate the independent variable(s).
• Control: Other variables are kept constant to isolate the effect of the independent variable.
• Randomization: Subjects are randomly assigned to different groups to minimize bias.
• Replication: The study can be replicated to verify results.

Types of Experimental Study

• Laboratory Experiments: Conducted in a controlled environment where variables can be precisely controlled.
• Field Research : These are conducted in a natural setting but still involve manipulation and control of variables.
• Clinical Trials: Used in medical research and the healthcare industry to test the efficacy of new treatments or drugs.

Example of an Experimental Study:

Imagine a study to test the effectiveness of a new drug for reducing blood pressure. Researchers would:

• Randomly assign participants to two groups: receiving the drug and receiving a placebo.
• Ensure that participants do not know their group (double-blind procedure).
• Measure blood pressure before and after the intervention.
• Compare the changes in blood pressure between the two groups to determine the drug’s effectiveness.

What is an Observational Study?

An observational study is a research design in which the investigator observes subjects and measures variables without intervening or manipulating the study environment. This type of study is often used when manipulating impractical or unethical variables.

Key Characteristics of Observational Studies:

• No Manipulation: Researchers do not manipulate the independent variable.
• Natural Setting: Observations are made in a natural environment.
• Causation Limitations: It is difficult to establish cause-and-effect relationships due to the need for more control over variables.
• Descriptive: Often used to describe characteristics or outcomes.

Types of Observational Studies: 

• Cohort Studies : Follow a control group of people over time to observe the development of outcomes.
• Case-Control Studies: Compare individuals with a specific outcome (cases) to those without (controls) to identify factors that might contribute to the outcome.
• Cross-Sectional Studies : Collect data from a population at a single point to analyze the prevalence of an outcome or characteristic.

Example of an Observational Study:

Consider a study examining the relationship between smoking and lung cancer. Researchers would:

• Identify a cohort of smokers and non-smokers.
• Follow both groups over time to record incidences of lung cancer.
• Analyze the data to observe any differences in cancer rates between smokers and non-smokers.

Difference Between Experimental vs Observational Studies

Choosing Between Experimental and Observational Studies

The researchers relied on statistical analysis to interpret the results of randomized controlled trials, building upon the foundations established by prior research.

Use Experimental Studies When:

• Causality is Important: If determining a cause-and-effect relationship is crucial, experimental studies are the way to go.
• Variables Can Be Controlled: When you can manipulate and control the variables in a lab or controlled setting, experimental studies are suitable.
• Randomization is Possible: When random assignment of subjects is feasible and ethical, experimental designs are appropriate.

Use Observational Studies When:

• Ethical Concerns Exist: If manipulating variables is unethical, such as exposing individuals to harmful substances, observational studies are necessary.
• Practical Constraints Apply: When experimental studies are impractical due to cost or logistics, observational studies can be a viable alternative.
• Natural Settings Are Required: If studying phenomena in their natural environment is essential, observational studies are the right choice.

Strengths and Limitations

Experimental studies.

• Establish Causality: Experimental studies can establish causal relationships between variables by controlling and using randomization.
• Control Over Confounding Variables: The controlled environment allows researchers to minimize the influence of external variables that might skew results.
• Repeatability: Experiments can often be repeated to verify results and ensure consistency.

Limitations:

• Ethical Concerns: Manipulating variables may be unethical in certain situations, such as exposing individuals to harmful conditions.
• Artificial Environment: The controlled setting may not reflect real-world conditions, potentially affecting the generalizability of results.
• Cost and Complexity: Experimental studies can be costly and logistically complex, especially with large sample sizes.

Observational Studies

• Real-World Insights: Observational studies provide valuable insights into how variables interact in natural settings.
• Ethical and Practical: These studies avoid ethical concerns associated with manipulation and can be more practical regarding cost and time.
• Diverse Applications: Observational studies can be used in various fields and situations where experiments are not feasible.
• Lack of Causality: It’s easier to establish causation with manipulation, and results are limited to identifying correlations.
• Potential for Confounding: Uncontrolled external variables may influence the results, leading to biased conclusions.
• Observer Bias: Researchers may unintentionally influence outcomes through their expectations or interpretations of data.

Examples in Various Fields

• Experimental Study: Clinical trials testing the effectiveness of a new drug against a placebo to determine its impact on patient recovery.
• Observational Study: Studying the dietary habits of different populations to identify potential links between nutrition and disease prevalence.
• Experimental Study: Conducting a lab experiment to test the effect of sleep deprivation on cognitive performance by controlling sleep hours and measuring test scores.
• Observational Study: Observing social interactions in a public setting to explore natural communication patterns without intervention.

Environmental Science

• Experimental Study: Testing the impact of a specific pollutant on plant growth in a controlled greenhouse setting.
• Observational Study: Monitoring wildlife populations in a natural habitat to assess the effects of climate change on species distribution.

How QuestionPro Research Can Help in Experimental vs Observational Studies

Choosing between experimental and observational studies is a critical decision that can significantly impact the outcomes and interpretations of a study. QuestionPro Research offers powerful tools and features that can enhance both types of studies, giving researchers the flexibility and capability to gather, analyze, and interpret data effectively.

Enhancing Experimental Studies with QuestionPro

Experimental studies require a high degree of control over variables, randomization, and, often, repeated trials to establish causal relationships. QuestionPro excels in facilitating these requirements through several key features:

• Survey Design and Distribution: With QuestionPro, researchers can design intricate surveys tailored to their experimental needs. The platform supports random assignment of participants to different groups, ensuring unbiased distribution and enhancing the study’s validity.
• Data Collection and Management: Real-time data collection and management tools allow researchers to monitor responses as they come in. This is crucial for experimental studies where data collection timing and sequence can impact the results.
• Advanced Analytics: QuestionPro offers robust analytical tools that can handle complex data sets, enabling researchers to conduct in-depth statistical analyses to determine the effects of the experimental interventions.

Supporting Observational Studies with QuestionPro

Observational studies involve gathering data without manipulating variables, focusing on natural settings and real-world scenarios. QuestionPro’s capabilities are well-suited for these studies as well:

• Customizable Surveys: Researchers can create detailed surveys to capture a wide range of observational data. QuestionPro’s customizable templates and question types allow for flexibility in capturing nuanced information.
• Mobile Data Collection: For field research, QuestionPro’s mobile app enables data collection on the go, making it easier to conduct studies in diverse settings without internet connectivity.
• Longitudinal Data Tracking: Observational studies often require data collection over extended periods. QuestionPro’s platform supports longitudinal studies, allowing researchers to track changes and trends.

Experimental and observational studies are essential tools in the researcher’s toolkit. Each serves a unique purpose and offers distinct advantages and limitations. By understanding their differences, researchers can choose the most appropriate study design for their specific objectives, ensuring their findings are valid and applicable to real-world situations.

Whether establishing causality through experimental studies or exploring correlations with observational research designs, the insights gained from these methodologies continue to shape our understanding of the world around us. 

Whether conducting experimental or observational studies, QuestionPro Research provides a comprehensive suite of tools that enhance research efficiency, accuracy, and depth. By leveraging its advanced features, researchers can ensure that their studies are well-designed, their data is robustly analyzed, and their conclusions are reliable and impactful.

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This week: the arXiv Accessibility Forum

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Computer Science > Computer Vision and Pattern Recognition

Title: cdm: a reliable metric for fair and accurate formula recognition evaluation.

Abstract: Formula recognition presents significant challenges due to the complicated structure and varied notation of mathematical expressions. Despite continuous advancements in formula recognition models, the evaluation metrics employed by these models, such as BLEU and Edit Distance, still exhibit notable limitations. They overlook the fact that the same formula has diverse representations and is highly sensitive to the distribution of training data, thereby causing the unfairness in formula recognition evaluation. To this end, we propose a Character Detection Matching (CDM) metric, ensuring the evaluation objectivity by designing a image-level rather than LaTex-level metric score. Specifically, CDM renders both the model-predicted LaTeX and the ground-truth LaTeX formulas into image-formatted formulas, then employs visual feature extraction and localization techniques for precise character-level matching, incorporating spatial position information. Such a spatially-aware and character-matching method offers a more accurate and equitable evaluation compared with previous BLEU and Edit Distance metrics that rely solely on text-based character matching. Experimentally, we evaluated various formula recognition models using CDM, BLEU, and ExpRate metrics. Their results demonstrate that the CDM aligns more closely with human evaluation standards and provides a fairer comparison across different models by eliminating discrepancies caused by diverse formula representations.

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Design, topology optimization, manufacturing and testing of a brake caliper made of scalmalloy ® for formula sae race cars.

1. Introduction

2. design of the braking caliper, 2.1. maximum braking force, 2.2. preliminary design, 2.3. oil channels and hydraulic seals, 2.4. topology optimization, 2.5. finite element analyses and manufacturing, 3. experimental tests, 3.1. strain gauge measurement with static disc, 3.2. dial gauge measurement with static disc, 3.3. strain gauge measurement with rotating disc, 4. conclusions, author contributions, data availability statement, acknowledgments, conflicts of interest.

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Vecchiato, L.; Capraro, F.; Meneghetti, G. Design, Topology Optimization, Manufacturing and Testing of a Brake Caliper MADE of Scalmalloy ® for Formula SAE Race Cars. Vehicles 2024 , 6 , 1591-1612. https://doi.org/10.3390/vehicles6030075

Vecchiato L, Capraro F, Meneghetti G. Design, Topology Optimization, Manufacturing and Testing of a Brake Caliper MADE of Scalmalloy ® for Formula SAE Race Cars. Vehicles . 2024; 6(3):1591-1612. https://doi.org/10.3390/vehicles6030075

Vecchiato, Luca, Federico Capraro, and Giovanni Meneghetti. 2024. "Design, Topology Optimization, Manufacturing and Testing of a Brake Caliper MADE of Scalmalloy ® for Formula SAE Race Cars" Vehicles 6, no. 3: 1591-1612. https://doi.org/10.3390/vehicles6030075

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Efficient reversible data hiding in encrypted images using Block Complexity and most significant bit inversion strategy

• Published: 02 September 2024

Cite this article

• Cheng-Hsing Yang 1 ,
• Chi-Yao Weng   ORCID: orcid.org/0000-0002-0619-1966 2 ,
• Chia-Ling Hung 1 &
• Shiuh-Jeng WANG 3  

Reversible data hiding in the encrypted images (RDHEI) has attracted more attention because RDHEI can be used for both information protection and image encryption. Many researches based on RDHEI have been proposed by using the Most Significant Bit (MSB) inversion to embed confidential information, but they might subject to errors when extracting the hidden information. This paper improves the approach based on MSB inversion and proposes a new RDHEI technique. Our approach hides the block’s position of the block in the image, which would cause misinterpretation in the original image, and then encrypts the image. The MSB inversion strategy is applied to embed the secret messages in the encrypted image. Since the location information of the error block is pre-hidden in the image, this information ensures that the secret message is correctly extracted and the image is fully recovered. We also created a multi-regular block complexity formula to determine the secret bits hidden in a block and recover the original block. In addition, we extended the design of four methods to cover various segmentation strategies and complexity calculation methods. According to the experimental results, our method can successfully extract the secret message and recover the original image intact after the encrypted image is embedded with the secret message. Generally, in using different image size, we averagely achieve the PSNR and embedding capacity of 39 experimental images at 40.633 dB and 46,298.46 bits, respectively.

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Acknowledgements

This work was partially supported by the National Science and Technology Council of the Republic of China under the Grant No. MOST 110-2221-E-153-002-MY2, MOST 111-2221-E153-005 and MSTC 112-2221-E-153-003.

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Cheng-Hsing Yang & Chia-Ling Hung

Department of Computer Science and Information Engineering, National Chiayi University, Chiayi, 600, Taiwan

Chi-Yao Weng

Department of Information Management, Central Police University, Taoyuan City, 333, Taiwan

Shiuh-Jeng WANG

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Yang, CH., Weng, CY., Hung, CL. et al. Efficient reversible data hiding in encrypted images using Block Complexity and most significant bit inversion strategy. Multimed Tools Appl (2024). https://doi.org/10.1007/s11042-024-20106-0

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Received : 20 October 2023

Revised : 09 July 2024

Accepted : 18 August 2024

Published : 02 September 2024

DOI : https://doi.org/10.1007/s11042-024-20106-0

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