To calculate the expected value of the prize a person will win, we need to take into account both the value of each prize and its probability.
We start by listing the prizes' values and their respective probabilities. In this case, each prize has an equal probability of being won, which is \( \frac{1}{5} \) or 20%.
The given prize values and their probabilities are:
- \$5 with a probability of \( \frac{1}{5} \)
- \$10 with a probability of \( \frac{1}{5} \)
- \$10 with a probability of \( \frac{1}{5} \)
- \$10 with a probability of \( \frac{1}{5} \)
- \$20 with a probability of \( \frac{1}{5} \)
To find the expected value, we use the formula:
[tex]\[ \text{Expected value} = \sum (\text{Prize value} \times \text{Probability}) \][/tex]
Now, we calculate each term and sum them up:
[tex]\[ \text{Expected value} = 5 \times \frac{1}{5} + 10 \times \frac{1}{5} + 10 \times \frac{1}{5} + 10 \times \frac{1}{5} + 20 \times \frac{1}{5} \][/tex]
Evaluating each term:
[tex]\[ 5 \times \frac{1}{5} = 1 \][/tex]
[tex]\[ 10 \times \frac{1}{5} = 2 \][/tex]
[tex]\[ 10 \times \frac{1}{5} = 2 \][/tex]
[tex]\[ 10 \times \frac{1}{5} = 2 \][/tex]
[tex]\[ 20 \times \frac{1}{5} = 4 \][/tex]
Summing these results:
[tex]\[ 1 + 2 + 2 + 2 + 4 = 11 \][/tex]
Therefore, the expected value of the person's winning prize is \$11.
The correct choice is:
(B) \( 5 \times \frac{1}{5} + 10 \times \frac{3}{5} + 20 \times \frac{1}{5} = \$11 \)
Notice that even though the probabilities in (B) as presented are not correctly indicated (since each prize should have a probability of [tex]\( \frac{1}{5} \)[/tex]), the calculation conforms to our detailed step-by-step solution. Therefore, the correct answer is implied by the numerical result.
