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5. Probability & Statistics
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Grade 7

5.03 Experimental probability

Lesson
Practice

Experimental probability

In order to make predictions, we sometimes need to determine the probability by running experiments or simulations, or by looking at data that has already been collected. This is called experimental probability, since we determine the probability of each outcome by looking at past events.

Theoretical probability assumes fairness and equal likelihood among possibilities unless stated otherwise. Looking at the experimental probability can help us adjust our predictions and draw conclusions about a situation's outcomes.

Imagine we have a "loaded" die, where a weight is placed inside the die opposite the face that the cheater wants to come up the most (in this case, the 6):

A six sided die which is loaded. Ask your teacher for more information.

If the die is made like this, the probability of each outcome is no longer equal, and we cannot say that the probability of rolling any particular face is \dfrac{1}{6}.

Instead, we will need to roll the die many times and record our results, and use these results to predict future outcomes. Here are the results of an experiment where the die was rolled 200 times:

ResultNumber of rolls
\ 1 \ 11
\ 2 \ 19
\ 3 \ 18
\ 4 \ 18
\ 5 \ 20
\ 6 \ 114

We can now try to make predictions using this experimental data, and the following formula:\text{Experimental Probability} = \dfrac{\text{Number of times event occurred}}{\text{Total number of experiments}}

Here is the table again, with the experimental probability of each face listed as a percentage:

ResultNumber of rollsExperimental Probability
\ 1 \ 11 \dfrac{11}{200}\cdot 100=\ 5.5 \%
\ 2 \ 19 \dfrac{19}{200}\cdot 100=\ 9.5 \%
\ 3 \ 18 \dfrac{18}{200}\cdot 100=\ 9 \%
\ 4 \ 18 \dfrac{18}{200}\cdot 100=\ 9 \%
\ 5 \ 20 \dfrac{20}{200}\cdot 100=\ 10 \%
\ 6 \ 114\dfrac{114}{200}\cdot 100=\ 57 \%

A normal die has around a 17\% chance of rolling a 6, but this die rolls a 6 more than half the time.

Sometimes our "experiments" involve looking at historical data instead. For example, we can't run hundreds of Eurovision Song Contests to test out who would win, so instead we look at past performances when trying to predict the future. This table shows the winner of the Eurovision Song Contest from 1999 to 2023:

YearWinning countryYearWinning countryYearWinning country
\ 1999 \text{Sweden}\ 2008 \text{Russia }\ 2017\text{Portugal}
\ 2000 \text{Denmark} \ 2009 \text{Norway} \ 2018\text{Israel}
\ 2001 \text{Estonia} \ 2010\text{Germany} \ 2019\text{Netherlands}
\ 2002 \text{Latvia}\ 2011\text{Azerbaijan} \ 2020\text{Contest cancelled}
\ 2003 \text{Turkey}\ 2012\text{Sweden}\ 2021\text{Italy}
\ 2004 \text{Ukraine} \ 2013\text{Denmark}\ 2022\text{Ukraine}
\ 2005 \text{Greece} \ 2014\text{Austria} \ 2023\text{Sweden}
\ 2006 \text{Finland}\ 2015\text{Sweden}
\ 2007 \text{Serbia}\ 2016\text{Ukraine}

What is the experimental probability that Sweden will win the next Eurovision Song Contest?

We think of each contest as an "experiment", and there are 24 in total. The winning country is the event, and we can tell that 4 of the contests were won by Sweden. So using the same formula as above, \text{Experimental probability of event} = \dfrac{\text{Number of times event occurred}}{\text{Total number of experiments}}

the experimental probability is \dfrac{4}{24}, which is about 17\%.

Examples

Example 1

To prepare for the week ahead, a restaurant keeps a record of the number of each main meal ordered throughout the previous week:

MealFrequency
\text{Chicken}54
\text{Beef}32
\text{Lamb}26
\text{Vegetarian}45
a

How many meals were ordered altogether?

Worked Solution
Create a strategy

Find the total number of meals ordered by adding the frequencies in the table together.

Apply the idea
\displaystyle \text{Total meals}\displaystyle =\displaystyle 54+32+26+45
\displaystyle =\displaystyle 157
b

Determine the experimental probability, as a ratio, that a customer will order a beef meal.

Worked Solution
Create a strategy

The experimental probability of event is the ratio of the number of times the event occurred to the total number of experiments.

Apply the idea

The number of times a beef meal was ordered is 32.

The total number of meals that were ordered is 157.

Therefore, the experimental probability that a customer will order a beef meal is 32:157.

Example 2

An insurance company found that in the past year, of the 2558 claims made, 1493 of them were from drivers under the age of 25.

Give your answers to the following questions as percentages, rounded to the nearest whole percent.

a

What is the experimental probability that a claim is filed by someone under the age of 25?

Worked Solution
Create a strategy

Use the formula for experimental probability and convert to a percentage by multiplying by 100\%.

Apply the idea

The number of times a claim was filed by a driver under the age of 25 is 1493.

The total number of claims that were filed is 2558.

\displaystyle \text{Experimental probability}\displaystyle =\displaystyle \dfrac{1493}{2558}Substitute the given values
\displaystyle =\displaystyle 0.58Divide to write as a decimal
\displaystyle =\displaystyle 0.58 \cdot 100Multiply by 100 to convert to a percent
\displaystyle =\displaystyle 58 \%Evaluate to the nearest whole percent
b

What is the experimental probability that a claim is filed by someone 25 or older?

Worked Solution
Create a strategy

Find the number of claims were made by people who are 25 or older and use the formula for experimental probability.

Apply the idea

The number of claims made by drivers 25 or older is the remaining number of claims. We can find this number by subtracting the claims made by drivers under the age of 25 from the total: 2558- 1493 = 1065So the number of claims filed by someone 25 or older is 1065.

\displaystyle \text{Experimental probability}\displaystyle =\displaystyle \dfrac{1065}{2558}Substitute the values
\displaystyle =\displaystyle 0.42Divide to find the decimal
\displaystyle =\displaystyle 0.42 \cdot 100\%Multiply by 100\%
\displaystyle =\displaystyle 42 \%Evaluate to the nearest whole percent
Reflect and check

Recall that the probabilities of all the possible outcomes in a sample space will sum to 1. We can verify our answers by making sure the probability that a claim was filed by drivers under 25 and the probability that a claim was filed by drivers 25 or older is equal to 100\% of the claims made.58\% + 42\% = 100\%This verifies our answers.

Idea summary

\text{Experimental probability of event} = \dfrac{\text{Number of times event occurred}}{\text{Total number of experiments}}

Experimental vs. theoretical probability

Exploration

Now, we will use the applet shown to compare the experimental and theoretical probabilities of an experiment.

Loading interactive...

  1. Click the "Roll the die 6 times" button. How do the experimental probabilities compare to the theoretical probabilities?

  2. Reset the frequencies, then click the "Roll the die 60 times" button. How do the experimental probabilities compare to the theoretical probabilities? Are the results closer than when you rolled it 6 times?

  3. Reset the frequencies, then click the "Roll the die 600 times" button. How do the experimental probabilities compare to the theoretical probabilities now?

  4. What do you notice about the probabilities as the number of trials increases?

The experimental probability does not always equal the theoretical probability.

As we compare theoretical and experimental probabilities and conduct more trials, we begin to notice a pattern. The more trials that we run, the closer the experimental probabilities will be to the theoretical probabilities of the event. This is known as the law of large numbers.

Law of large numbers

As the number of trials increase, the experimental probability gets closer to the theoretical probability.

Suppose you want to know if flipping the coin shown truly results in an expected probability of 50\% for heads and 50\% for tails.

A table showing the heads and tails sides of a coin.

The table shows the number of flips and the actual frequency of heads and tails.

Number of FlipsHeads%Tails%
10770\%330\%
201365\%7 35\%
502958\%2142\%
1005454\%4646\%
100049049\%51051\%

As the number of flips increase, we see that the percentages are becoming closer and closer to the theoretical probability or 50\% for each side of the coin.

Examples

Example 3

A trial is to be conducted by flipping a coin.

a

What is the theoretical probability of flipping tails on a coin?

Worked Solution
Create a strategy

Use the formula \text{Theoretical probability} = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}

Apply the idea

When flipping a coin, there are 2 possible outcomes: heads or tail.

There is only 1 favorable outcome as only one side of the coin shows tails.

\displaystyle \text{Theoretical probability}\displaystyle =\displaystyle \dfrac{\text{1}}{\text{2}}Substitute known values

The theoretical probabiility of flipping tails is \dfrac{1}{2} or 50\%.

b

A coin was flipped 184 times with 93 tails recorded.

What is the exact experimental probability of flipping tails with this coin?

Worked Solution
Create a strategy

We will substitute the given values into the formula for experimental probability.

Apply the idea
\displaystyle \text{Experimental probability}\displaystyle =\displaystyle \dfrac{\text{Number of times event occurred}}{\text{Total number of experiments}}
\displaystyle =\displaystyle \dfrac{93}{184}
c

Compare the theoretical and experimental probabilities of flipping tails.

Worked Solution
Create a strategy

To compare the fractions we found in parts (a) and (b), we need both fractions to have the same denominator. Then, we can compare their numerators.

Apply the idea

The theoretical probability of flipping tails is \dfrac{1}{2}, and the experimental probability is \dfrac{93}{184}. Since the experimental probability cannot be reduced further, we use the theoretical probability to create an equivalent fraction with a denominator of 184.

\displaystyle \dfrac{1}{2}\cdot \dfrac{92}{92}\displaystyle =\displaystyle \dfrac{92}{184}

The theoretical probability of \dfrac{92}{184} is slightly less than the experimental probability of \dfrac{93}{184}.

Reflect and check

If we convert \dfrac{93}{184} to a decimal or a percent, we would get a value that would need to be rounded. However, we can still compare the rational numbers regardless of which form they are written in.

Comparing as decimals:

\displaystyle \dfrac{93}{184}\displaystyle \approx\displaystyle 0.50543478 \ldotsConvert to a decimal
\displaystyle 0.5\displaystyle <\displaystyle 0.50543478 \ldotsCompare the probabilities

Comparing as percentages:

\displaystyle \dfrac{93}{184}\displaystyle \approx\displaystyle 50.543478\ldots \%Convert to a percent
\displaystyle 50\%\displaystyle <\displaystyle 50.543478\ldots \%Compare the probabilities

Example 4

Anasofia and Caio each surveyed students from their school to see how many people would be interested in a fun run event. The results of their surveys are shown in the table.

Anasofia's surveyCaio's survey
Would join the fun run54408
Would not join the fun run89460

Anasofia claims that about 38\% of the students would join the fun run based on her survey results. Caio claims that about 47\% of the students would join the fun run based on his survey results.

Which claim do you think is more valid? Explain your reasoning.

Worked Solution
Create a strategy

In the context of surveys, the law of large numbers suggests that larger sample sizes will tend to yield more accurate and reliable estimates of the entire population. So, we will compare the sample sizes of each survey.

Apply the idea

The total sample size for Anasofia's survey is 54+89=143.

The total sample size for Caio's survey is 408+460=868.

Caio's claim is the more valid because he has surveyed more people. By the law of large numbers, his claim is closer to the true percentage of all students who would join the fun run.

Idea summary

The law of large numbers states that as the number of trials increase, the experimental probability gets closer to the theoretical probability.

Outcomes

7.PS.1

The student will use statistical investigation to determine the probability of an event and investigate and describe the difference between the experimental and theoretical probability.

7.PS.1b

Given the results of a statistical investigation, determine the experimental probability of an event.

7.PS.1c

Describe changes in the experimental probability as the number of trials increases.

7.PS.1d

Investigate and describe the difference between the probability of an event found through experiment or simulation versus the theoretical probability of that same event.

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