# Video: Calculating Expected Values

The table shows the number of cars that 65 families have. Find the mean number of cars per family. The data can be expressed as a probability distribution for the discrete random variable π as shown. Find the value of π, π, π, and π. Calculate the expected value of π.

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### Video Transcript

The table shows the number of cars that 65 families have. Find the mean number of cars per family.

To find the mean from a frequency table, we can apply the given formula. In this case, π is the frequency and π₯ is the number of cars each family owns. Letβs begin by finding the sum of the products of the number of cars and their frequency.

For the first column, thatβs one multiplied by 10. We then have 35 families who own two cars, so thatβs two multiplied by 35; 15 families who own three cars β thatβs three multiplied by 15; and for the final column, thatβs four multiplied by five. Thatβs 10 plus 70 plus 45 plus 20, which is equal to 145.

Now we donβt actually need to work out the total frequency, but if we did, we could just add 10, 35, 15, and five. In fact, weβre told that there are 65 families, so we can simply divide 145 by 65. 145 divided by 65 is twenty-nine thirteenths.

We can check whether the answer is likely to be correct by considering the context of our question. Twenty-nine thirteenths is a little over two, and the number of cars per family were either one, two, three, or four. So twenty-nine thirteenths is a sensible answer for the mean number of cars per family.

The data can be expressed as a probability distribution for the discrete random variable π as shown. Find the value of π, π, π, and π.

To answer this, weβll need to consider the relative frequency for each value of π. When the number of cars per family is one, π₯ is one. There were 10 families who had one car and a total of 65 families in this study. This means the relative frequency or the probability that a person chosen at random from this group had one car is 10 over 65, which simplifies to two thirteenths, so π is two thirteenths.

We can repeat this process to find the values of π, π, and π. π is the probability that a family chosen at random has two cars. There were 35 families who had two cars, so the probability that a family chosen at random has two cars is 35 out of 65, which simplifies to seven thirteenths. There were 15 families who had three cars, so the probability a person chosen at random had three cars is 15 over 65. That means that π is three thirteenths. And finally, π represents the probability that a family chosen at random has four cars. There were five families who had four cars, so the probability of choosing a family at random with four cars is five over 65, which is one thirteenth. π is two thirteenths, π is seven thirteenths, π is three thirteenths, and π is one thirteenth.

Calculate the expected value of π.

To find the expected value of a discrete random variable π, we find the sum of the products of π multiplied by the probability of π occurring. In this case, thatβs one multiplied by two thirteenths, two multiplied by seven thirteenths, three multiplied by three thirteenths, and four multiplied by one thirteenth, which is twenty-nine thirteenths.

Notice the expected value of π is simply the mean, so we couldβve used the answer from the first part of this question to find the expected value of π. Either way, the expected value of π is twenty-nine thirteenths.