# Lesson Explainer: Experimental Probability Mathematics

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

Calculating the probability of an event is determining the likelihood that this event will occur. There are two main ways in which we can estimate the probability of an event. One way is to consider the attributes or physical properties of the event in question. For example, if we wanted to calculate the probability of rolling a 5 on a fair die, we would consider the number of sides on the die. This probability, , would be the theoretical probability of rolling a 5.

Often, however, we cannot use theoretical probability, for example, if we wanted to determine the probability of rolling a 5 on an unfair die. In this case, we would need to carry out an experiment where we roll the die a number of times, for example, 100 times, and record the number of times a 5 was rolled. This is termed experimental probability. To gather data to calculate experimental probability, we perform repeated trials and record the outcome (the result) of each trial.

### Definition: Experimental Probability

The experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials in an experiment.

Let’s consider an example where we have a biased (unfair) spinner with values from 1 to 6 that has some of its sides weighted. We want to determine the probability of the spinner landing on 2.

As the spinner is biased, we cannot use the fact that it has 6 equal sides to determine the probability of getting any one of the values. We would need to perform an experiment where we repeatedly spin the spinner a number of times and record the outcome of each spin. A tally chart or table is useful for recording these results. The more trials that we perform, the more reliable our results will be. However, we must also balance this with the practical considerations of time and cost.

Let’s say that after the experiment is complete, we have the table of outcomes below.

 Value on Spinner Number of Outcomes 1 2 3 4 5 6 2 16 25 17 4 0

There are a number of things we can observe from the table, for instance, the value 3 on the spinner occurred most times: it had a frequency of 25 spins. In comparison, the value 6 was never spun on the spinner during the experiment: its frequency was 0.

From a table such as this, we can also determine the number of trials that were carried out in the experiment: we would calculate the total number of outcomes. This would give

Importantly, we can also use the results of this experiment to find the experimental probability of spinning each or any value on the spinner. In general, to calculate experimental probability, we have

We can reword this formula to match the context of the probability that we need to calculate. Here, we could write the experimental probability of spinning a 2 as

Given that the number of trials in which 2 occurs is 16 and the total number of trials is 60, we have

We can write probabilities as fractions, decimals, or percentages, so an answer of , 0.25, or would be valid.

Note that in a context where we are asked to find the probability of an event from data given as a set of results, a table, or a graph, this will be the experimental probability.

Let’s now see how we can apply this in the following examples.

### Example 1: Calculating the Experimental Probability of an Event Using a Frequency Table

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

 Breakfast Number of Students Eggs Cereal Toast 10 2 8

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

### Answer

In the table above, we can see that out of 20 students, 10 students said that eggs were their favorite breakfast, 2 students said that cereal was their favorite, and 8 students said that toast was their favorite. In order to calculate the probability that a student prefers eggs, we will calculate the experimental probability of this. In general, we use the formula

In this context, the experimental probability of a randomly selected student preferring eggs can be written as

From the table, we have the number of students surveyed who preferred eggs as 10, and we are given that the total number of students surveyed is 20. Even if we did not have the information about the total number of students being 20, we could calculate this by summing the number of students preferring eggs (10), cereal (2), and toast (8).

Hence, we have

Therefore, we can give the answer that the probability that a randomly selected student prefers eggs is

Let’s see another example where the outcomes of an experiment are given as a list, or set, of results.

### Example 2: Calculating the Experimental Probability of an Event Using a Set

Dina creates a three-sided spinner using the colors red, green, and blue. She spins the spinner and records the following results:

{red, blue, red, green, green, green, red, red, red, green}.

Calculate the experimental probability of spinning green on this spinner.

### Answer

The experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials in an experiment. It can be written as

In this problem, we will calculate

Using the given set of results, we can determine that green was spun 4 times, as it appears 4 times in the list of results. By counting all the results, we determine that the total number of spins in the experiment is 10, which is the size of the set of results. Thus, we have

Probability can be given as a fraction, decimal, or percentage, so , 0.4, or are valid answers here.

We can give the answer that the experimental probability of spinning green on this spinner is

Note that in this question, the spinner was spun 10 times. Ideally, a larger number of spins would give a more accurate result.

Experimental probability is particularly useful in allowing us to determine the probability of an event happening on a wider scale by performing an experiment on a smaller scale. This is often done in industry by carrying out experiments on a sample. For example, a manufacturer of light bulbs might test a sample of light bulbs to determine the length of their lifetime. Since this would destroy the bulbs that are tested, the manufacturer cannot perform this check on each of the bulbs the manufacturer has produced. Rather, selecting 100 light bulbs and determining that 97 of them reached a target time of 500 hours would allow the manufacturer to extrapolate that 0.97 of the manufactured bulbs have a lifetime of at least 500 hours.

We will now see an example of finding the experimental probability of an event from a sample.

### Example 3: Estimating the Probability of an Event Using Experimental Data

A store receives a box of apples from an orchard. A worker inspects a sample of 54 apples from the box. Of these apples, 6 are spoiled. Use this data to estimate the probability that an apple received from the orchard is spoiled.

### Answer

We can estimate the probability of the store receiving a spoiled apple by applying experimental probability. In this context, the experiment is that of repeated trials of selecting apples. The two different outcomes are selecting a spoiled apple or selecting an unspoiled apple.

We can calculate the probability of selecting a spoiled apple using the number of spoiled apples (6) and the total number of apples (54) as

Hence, we can give the answer that the estimated probability of receiving a spoiled apple is

The way in which the data from experimental probability is presented can be in the form of a statement, a set of results, a table, or a graph. We still apply the same process of calculating the experimental probability of an event by dividing the number of trials in which that particular outcome occurs by the total number of trials. In many cases, we may need to sum the frequencies of the different outcomes to find the total number of trials.

In the following question, we will see how we can use a bar graph to calculate an experimental probability.

### Example 4: Calculating the Experimental Probability of an Event Using a Graph

The graph shows the results of an experiment in which a die was rolled 26 times. Find the experimental probability of rolling a 2. Give your answer as a fraction in its simplest form.

### Answer

In this question, we are given a bar graph representing the number of times the values 1 to 6 were rolled on a die. We can calculate the experimental probability of rolling a 2 as

Using the bar graph, we can observe that the number of times 2 was rolled is 8.

We are given that the total number of rolls is 26. We can verify this by finding the total of all the frequencies (the total number of rolls) of the values 1 to 6. This is given by . Note that we must also include the number of times 2 was rolled (8) in our sum.

We now calculate the experimental probability of rolling a 2 as

Therefore, the experimental probability of rolling a 2 is

We will now see another question.

### Example 5: Calculating the Experimental Probability of an Event

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 this game, we can observe that there are, in fact, 4 different outcomes: winning gold, silver, bronze, or not winning any of these prizes.

Given the information that 3 people won the gold prize, 12 won the silver prize, and 15 won the bronze prize out of a total of 68 participants, we can calculate the number of participants who won no prizes as

We can then calculate the experimental probability of not winning any prizes by dividing the number of participants who won no prizes by the total number of participants, that is,

Thus, we have

Therefore, we can give the answer that the experimental probability of not winning any prizes in this game is

In the final question, we will use a given experimental probability and a value for the number of outcomes to calculate the total number of trials in an experiment.

### Example 6: Determining the Total Number of Trials in an Experiment

The experimental probability that a coin lands on tails is . If the coin landed on tails 30 times, how many times was it tossed in the experiment?

### Answer

In this question, we are given the experimental probability that a coin lands on tails. This probability has been calculated using the data from an experiment with repeated trials of tossing a coin. We are also given that the number of outcomes of landing on tails (the number of times the coin landed on tails) is 30. We recall that in general, experimental probability is calculated as

In this context, we would have

Given that the probability of the coin landing on tails is and the coin landed on tails 30 times, we have

Then, by cross multiplying, we have

Dividing both sides by 3 and simplifying gives us

Therefore, we can give the answer that the coin was tossed 70 times.

We now summarize the key points of this explainer.

### Key Points

• Experiments can be used to estimate the probability of an event occurring.
• To gather data to calculate experimental probability, we perform repeated trials and record the outcome (the result) of each trial.
• The more trials we perform, the more accurate an estimate of results we will get. However, in a real-life context, we must balance this with the time and cost considerations of performing a large number of trials.
• The experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials in an experiment.
• We can calculate experimental probability from data presented in the form of a statement, set of results, a table, or a graph.

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