Explainer: Experimental Probability

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: