In this explainer, we will learn how to predict the actions of a larger group using a sample and find the expected value.
When we talk of the probability of something happening, we mean the chance or likelihood of it occurring. The probability of an event tells us how likely something is to happen in the long run, and there are two ways of looking at this:
- Experimental probability is calculated by looking at the outcomes of a repeated experiment. Experimental probability is also sometimes called the relative frequency.
- Theoretical probability is calculated using mathematical reasoning about the possible outcomes.
Theoretically, we know that if we toss a fair coin, we have a (or a 1 in 2) chance of getting heads; this is the theoretical probability. If stands for heads, we write ; that is, the probability of getting heads is 0.5.
If we were to toss a coin 100 times, however, we might find that we get heads on only 45 out of the 100 throws. So in this case, the experimental probability of heads is , or , which is as a percentage. Clearly, this value is less than our theoretical probability.
The expected value is what we expect to happen on average, based on either theoretical or experimental probabilities, if we were to perform many trials or experiments.
Formula: Expected Value
The expected value is calculated by multiplying the probability of the event occurring by the number of times the experiment is performed; that is,
Let’s see how the expected value works in some examples.
Example 1: Calculating the Expected Value of an Event
The probability that a biased die will land on an even number is 0.6. If the die is rolled 80 times, how many times is it expected to land on an even number?
Answer
First, recall the formula for the expected value:
In this case, the event is that the biased die lands on an even number when rolled, which has a probability of 0.6. The experiment is rolling the die, which is performed 80 times. Substituting into the formula, we have
Therefore, if this biased die is rolled 80 times, the number of times we would expect it to land on an even number is 48.
In the above example, we were given both the probability of the event and the number of times the experiment was performed, and we substituted these numbers directly into the expected value formula. Sometimes, however, we are given a data set and must first work out the experimental probability of the event from the data before we can apply the formula.
Example 2: Making Inferences from a Data Set
In a survey of 400 tourists who visited Egypt, 160 were from Arab countries, 120 were from Europe, 40 from Latin America, and 80 from Australia. If the total number of tourists who visited Egypt in a month was 5 000, how many of them are expected to be from Europe?
Answer
Recall the formula for the expected value:
From the question, the experiment is a tourist visit to Egypt, which occurred 5 000 times in a month. The event is that the tourist was from Europe.
Before we can find the expected number of tourists out of 5 000 who were from Europe, we must first use the survey data to work out the experimental probability that a tourist was from Europe. The probability is classed as experimental, as opposed to theoretical, since the data comes from a survey.
There were 400 tourists surveyed, of which 120 were from Europe. Therefore, the probability that a tourist chosen at random from this survey was from Europe is
Now that we have worked out the experimental probability of the event we are interested in, we can apply the expected value formula as follows. We have a total of 5 000 tourists visiting Egypt in the given month, so
Hence, we have shown that out of 5 000 tourists visiting Egypt in the given month, we would expect 1 500 of them to be from Europe.
Our next example features the results of an experiment. From this, we can work out the probability of the event we are interested in and then use it to make predictions about the expected value if we had performed the experiment many more times.
Example 3: Calculating an Expected Value from an Experiment
The table shows the results of rolling a die 78 times. Using this information, how many times is the number 2 expected to appear if the die is rolled 234 times?
Answer
Recall the formula for the expected value:
Before we can apply this formula to calculate how many times the number 2 is expected to appear if the die is rolled 234 times, we first need to use information from the table to work out the experimental probability of throwing a 2.
The first (left-hand) column of the table tells us the number on the face of the die, and the second column tells us how many times a numbered face occurred out of 78 rolls. This means, for example, that the number 2 occurred 17 times out of 78 rolls of the die.
The probability of rolling a 2 with this die is therefore which is 0.218 to three decimal places. To work out the expected number of 2s from 234 rolls, we now apply the expected value formula:
The expected number of 2s from 234 rolls of this die is therefore 51.
Note that in the above example, we used the experimental probability, which we worked out from observed data: out of 78 throws of the die, 17 landed with the number 2 face up. This told us that the probability of throwing a 2 with this die is . If we had used the theoretical probability, we would have had a different result.
Theoretically, each face on a die should have an equal probability of occurring. That is, the probability for each face should be the same, . If we had used this probability, then out of 234 throws, we would expect to get a 2 a total of times.
In some questions, we are given details of a random sample from a specific population. In such cases, we need to use information about the sample to calculate the probability of the event we are interested in. This result can then be used to make predictions about the wider population from which the random sample was taken.
Example 4: Calculating an Expected Value given a Sample of Data
A random sample of 80 school students were asked to vote for their preferred sport and the results are shown in the table. If the whole school contains 700 students, how many are expected to prefer football?
| Game | Football | Basketball | Table tennis | Swimming |
|---|---|---|---|---|
| Number of Students | 32 | 24 | 8 | 16 |
Answer
Recall the formula
Before we can apply this formula to calculate how many students from a school of 700 pupils are expected to prefer football, we need to use information about the sample to work out the probability that a randomly chosen student prefers football.
From the table, we have that 32 students out of the sample of 80 prefer football. This means that the experimental probability of a student preferring football is
We can now use this probability to calculate the expected number of students in the whole school that prefer football. Applying the expected value formula, we have
We conclude that from this school of 700 students, 280 of them are expected to prefer football.
In our final example, the probability of the event we are interested in is derived from a sample and expressed in percentage form; this must be converted to a fraction or decimal before we can use it in the expected value formula.
Example 5: Finding the Expected Number of Defective Items Produced by a Factory
A factory produced 1 600 calculators in one day. They took a sample of those calculators and found that were defective. What is the expected number of defective calculators produced that day?
Answer
Recall the expected value formula:
In this question, the experiment is the production of calculators in a factory, which occurs 1 600 times in one day. The event we are interested in is that a calculator is defective.
Before we can apply the formula, we first need to use information about the sample to work out the probability of a randomly chosen calculator being defective. Note that we are not given the number of calculators involved in the sample, but this does not matter because we are told that of them were defective; in other words, 3 out of every 100 were defective. Therefore, the probability that a calculator from this sample was defective is
This probability can now be substituted into the expected value formula, so we have
We deduce that the expected number of defective calculators produced that day is 48.
Let’s finish by recapping some key concepts from this explainer.
Key Points
- The expected value is what we expect to happen on average over many trials or experiments. It is calculated by multiplying the probability of the event occurring by the number of times the experiment is performed:
- We can often calculate the probability from data, such as the results of a survey or sample. In cases like this, to find the expected value, we take the experimental probability we have calculated and multiply it by the total number of trials.
