Multiply the amount of money on the gift cards by their corresponding probabilities. Then, add all of the products together.
C
Practice makes perfect
The expected value of a discrete, random variable is the weighted sum of its values. In our case, the values of the random variable represent the amount of money on the gift cards that can be won in a mall contest.
E(Money Per Card)= ∑ [Money* P(Money)]
In other words, to find the expected value of the gift card, we need to add up all the products of all the possible amounts of money on the gift cards and their corresponding probabilities. Therefore, we first need to convert frequencies, which are the given number of winners, to probabilities. Let's begin by finding the total number of winners.
495+405+285+180+90+45= 1500
To find the probabilities, we need to divide each number of winners of each value of the gift card by 1500, which is the obtained total number of winners.
Amount, X
Winners
Probability
$ 100
495
495/1500 = 0.33
$ 125
405
405/1500 = 0.27
$ 150
285
285/1500 = 0.19
$ 200
180
180/1500 = 0.12
$ 250
90
90/1500 = 0.06
$ 300
45
45/1500 = 0.03
Now we are ready to find the products of the possible amounts of money on the gift cards and their corresponding probabilities.
Amount, X
Probability
Money * P(Money)
$ 100
495/1500 = 0.33
100 * 0.33 = 33
$ 125
405/1500 = 0.27
125 * 0.27 = 33.75
$ 150
285/1500 = 0.19
150 * 0.19 = 28.5
$ 200
180/1500 = 0.12
200 * 0.12 = 24
$ 250
90/1500 = 0.06
250 * 0.06 = 15
$ 300
45/1500 = 0.03
300 * 0.03 = 9
We can calculate the expected value by adding all values from the last column. Let's do it!
E(Money per Card)&=
&=33 + 33.75 + 28.5 + 24 + 15 + 9
&=143.25
The expected value of the gift card that is won is $ 143.25. This corresponds to answer C.
