Lesson Explainer: Experimental Probability Mathematics • 7th Grade

In this explainer, we will learn how to interpret a data set by finding and evaluating the experimental probability.

Probability refers to the likelihood or chance of an event occurring. In experimental probability, we make estimates for the likelihood or chance of something occurring based on the results of a number of experiments or trials.

Let us look at an example to illustrate what we mean by this. We will calculate the experimental probability using data in a table.

Example 1: Experimental Probability from a Table

The table shows the results of a survey that asked 20 students about their favorite breakfast.

What is the probability that a randomly selected student prefers eggs?

Answer

As we are using data collected in a survey, this is classified as experimental data. The 20 students asked about their favorite breakfast are the 20 trials in our experiment.

Since 10 out of the 20 students prefer eggs, the probability that a randomly selected student prefers eggs is probabilitythatastudentpreferseggsEggsnumberofstudentswhoprefereggstotalnumberofstudents=𝑃()==1020=12=0.5.

The probability that a student selected at random prefers eggs for breakfast is therefore 0.5. Or converting this to a percentage, we can say there is a 50% chance that a student selected at random prefers eggs for breakfast. (Multiplying the probability 0.5 by 100 gives us the 50%.)

We will look at some more examples, but before we do this let us list the main points of interest for experimental probability.

Experimental Probability: Main Points

The experimental probability of event 𝐸 is an estimate of the probability for the event 𝑃(𝐸), based on data from a number of trials or experiments. So, for example, if we use data collected in a survey to estimate a probability, this would be classed as experimental probability.

  • The experimental probability of an event is often also called the relative frequency of the event and is given by relativefrequencyofeventnumberoftimesoccurstotalnumberoftrials𝐸=𝑃(𝐸)=𝐸.
  • As with any probability, if we have the experimental probability of an event 𝐸, we can find the probability that 𝐸 does not occur, 𝑃(𝐸), by using the total probability rule: thesumoftheprobabilitiesforallpossibleoutcomes=1. So 𝑃(𝐸)=1βˆ’π‘ƒ(𝐸).

The next example demonstrates how to calculate the relative frequency of an event.

Example 2: Experimental Probability and Relative Frequency

A coin was tossed 200 times and the number of tails observed was 102. Calculate the relative frequency of getting a heads. Calculate the answer to three decimal places.

Answer

As the coin was tossed 200 times, this means there were 200 β€œtrials.” We are looking for the relative frequency of getting a heads, but we know that 102 of the throws resulted in tails. So to begin, we will calculate the relative frequency of getting a tails, which is relativefrequencyofgettingatailsTailsnumberoftimestailsoccurredtotalnumberoftrials=𝑃()==102200.

Hence, the relative frequency of getting a tails is 102200. We can use this to find the relative frequency of getting a heads, which can be worked out in two different ways with the information we have.

  • Method 1 uses the rule that the sum of the probabilities for all possible outcomes is equal to 1. We have just worked out the relative frequency (or probability) of getting a tails. This is 102200. Since there is only one other possible outcome (heads), subtracting the β€œtails” probability from 1 gives us the probability (or relative frequency) of getting a head: probabilityofheadsprobabilityoftails=1βˆ’=1βˆ’102200=98200. To three decimal places, 98200=0.490.
  • Method 2 uses the number of tails that occurred and the total number of trials to calculate directly the number of heads out of all the trials. We then use this to calculate the relative frequency or probability of a heads: numberofheadsnumberoftrialsnumberoftails=βˆ’=200βˆ’102=98. As there were 98 heads out of 200 trials, the relative frequency of getting a heads is relativefrequencyofheadsHeadsnumberoftimesβ€œheads”occurredtotalnumberoftrials=𝑃()==98200=0.49. The relative frequency of getting a heads is therefore 0.490 to 3 decimal places.

In our next example, we calculate the experimental probability for an event.

Example 3: Experimental Probability

A game at a festival challenged people to throw a baseball through a tire. Of the first 68 participants, 3 people won the gold prize, 12 won the silver prize, and 15 won the bronze prize.

What is the experimental probability of not winning any of the three prizes?

Answer

In order to find the experimental probability of not winning any of the three prizes, let us first summarize the information that we have:

The total number of trials is the total number of participants, which is 68. We know also that, of these 68 participants, 30 won a prize. If 30 won a prize, then 68βˆ’30=38 participants did not win a prize and we can use this information to calculate the experimental probability of not winning any of the three prizes: probabilityofnotwinningaprizenumberofnon-prizewinnerstotalnumberofparticipants==3868=1934.

Hence, the experimental probability of not winning a prize is 1934.

Note that we could also have used the total probability rule to answer this question (i.e., that the sum of the probabilities for all possible outcomes is equal to 1). With 68 participants, 30 of whom won a prize, the probability of winning a prize is 3068=1534. By the total probability rule, the probability of not winning a prize is 1βˆ’π‘ƒ()Winning. So probabilityofwinning(probabilityofwinning)asbeforenot=1βˆ’=1βˆ’1534=1934.

Our next example demonstrates the use of tabular information in finding relative frequencies.

Example 4: Using Information from a Table to Find Relative Frequencies

The table shows the music preferences of a group of men and women.

  1. Calculate the relative frequency of a randomly selected person being a woman who prefers country music. If necessary, round your answer to 3 decimal places.
  2. Calculate the relative frequency of a randomly selected woman preferring rock music. If necessary, round your answer to 3 decimal places.

Answer

In order to find the relative frequencies, our first step is to calculate the total number of trials. We can work out the totals for each category as in the table below.

Part 1

To calculate the relative frequency of a randomly selected person being a woman who prefers country music, we must find the number of people falling into this category and also the total number of people. (The total number of people is the number of trials.)

We can see that the number of women who prefer country music is 13 and that the total number of people whose preferences were recorded (i.e., the number of trials) is 63. The relative frequency (or probability 𝑃) of a randomly selected person being a woman (W) who prefers country music (C) is, therefore, 𝑃()==1363=0.206.WandCnumberofwomenwhoprefercountrymusictotalnumberofpeopleto3d.p

The relative frequency, to three decimal places, of randomly selecting a woman who prefers country music is therefore 0.206. Or as a percentage, there is a 20.6% (=0.206Γ—100) chance that a person selected at random is a woman and prefers country music.

Part 2

To calculate the relative frequency of a randomly selected woman preferring rock music, we need to know how many women there were in total and the number of those who prefer rock music.

There was a total of 37 women, 24 of whom prefer rock music. So the relative frequency of women who prefer rock music is relativefrequencyofwomenwhopreferrockwomenrockerstotalwomento3d.p==2437=0.649.

Hence, the relative frequency of a randomly selected woman preferring rock music is 0.649 to three decimal places. Or, as a percentage, there is approximately a 65% (β‰ˆ0.649Γ—100) chance that a woman selected at random prefers rock music.

Note that there is a subtle difference between the wording of the two parts of this question. Part 1 asks for the relative frequency of a β€œrandomly selected person being a woman who prefers country music.” And part 2 refers to a β€œrandomly selected woman preferring rock music.”

The distinction is that, in part 2, we are making a random selection only from the women, whereas in part 1 we are randomly selecting from all of the people whose preferences were recorded. That is why in the solution to part 2 our denominator is the total number of women only, and in part 1 the denominator is the overall total, that is, both men and women.

Sometimes when looking at experimental probability, we may not be able to directly perform the experiment we would like but it is possible to model the situation. In the next example, we will consider how an experiment might be constructed to calculate experimental probability.

Example 5: Constructing an Experiment for Experimental Probability

One out of every six students in a seventh-grade class is left handed. Which of the following could be used to find the experimental probability that we will get a left-handed student when choosing randomly?

  1. Using a spinner with four colors: red represents right handed, and yellow, blue, and green represent left handed.
  2. Using a coin: heads is right handed, and tails is left handed.
  3. Using a number cube: even numbers represent right handed and odd numbers represent left handed.
  4. Using a number cube: landing on 1 represents left handed, and landing on 2–6 represents right handed.

Answer

Our information is that one in every six students is left handed. This means that for every individual left-handed student we expect there to be 5 right-handed students. So we are looking for an experiment with this ratio in its design. Let us consider each of the options separately and see if they fit the bill.

  1. Using a spinner with four colors: red represents right handed, and yellow, blue, and green represent left handed.
    This option will not fit. The spinner has only four colors and the ratio specified in this option is 1 right handed to 3 left handed, so there is no possibility that we can fit the ratio 1 left handed to 5 right handed to this scenario.
  2. Using a coin: heads is right handed, and tails is left handed.
    This option does not fit the bill either. The coin has only two possible outcomes, in the ratio 1 to 1. So we cannot fit our ratio 1 to 5 in this scenario.
  3. Using a number cube: even numbers represent right handed and odd numbers represent left handed.
    Using a number cube looks promising as it has 6 faces; however, the specification that β€œeven numbers represent right-handed and odd numbers represent left-handed students” does not work for us. As there are equal numbers of odd and even faces, this has reduced our ratio again to 1 to 1. For every even face, there is an odd face.
  4. Using a number cube: landing on 1 represents left handed, and landing on 2–6 represents right handed.
    In this case, we have a number cube where 1 represents left handed and the numbers 2–6 represent right handed. This works for us since the ratio of possibilities is 1 to 5. From 2 to 6 (including 2 and 6), there are 5 numbers. So this scenario could be used to find the experimental probability that we will get a left-handed student when choosing randomly.

Let us remind ourselves of the key ideas related to experimental probability

Key Points

  • The experimental probability of event 𝐸 is an estimate of the probability for the event 𝑃(𝐸), based on data from a number of trials or experiments.
  • The experimental probability of an event is also called the relative frequency of the event and is given by relativefrequencyofeventnumberoftimesoccurstotalnumberoftrials𝐸=𝑃(𝐸)=𝐸.
  • When calculating probabilities, we often use the β€œtotal probability rule”: thesumoftheprobabilitiesforallpossibleoutcomes=1.

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