To determine the expected value of the person's winning prize, we need to use the concept of expected value in probability. The expected value [tex]\(E[X]\)[/tex] can be calculated using the formula:
[tex]\[ E[X] = \sum (x_i \times P(x_i)) \][/tex]
where [tex]\(x_i\)[/tex] represents the cash value of each prize and [tex]\(P(x_i)\)[/tex] is the probability of winning that prize.
Given the prizes and their values: \[tex]$5, \$[/tex]10, \[tex]$10, \$[/tex]10, and \[tex]$20, we can calculate the probability of winning each prize. There are 5 prizes, so the probability of winning any one specific prize is:
- Probability of winning \$[/tex]5: [tex]\(P(\$5) = \frac{1}{5}\)[/tex]
- Probability of winning \[tex]$10: \(P(\$[/tex]10) = \frac{3}{5}\) (since there are three \[tex]$10 prizes)
- Probability of winning \$[/tex]20: [tex]\(P(\$20) = \frac{1}{5}\)[/tex]
Now, we calculate the expected value [tex]\(E[X]\)[/tex]:
[tex]\[ E[X] = 5 \times \frac{1}{5} + 10 \times \frac{3}{5} + 20 \times \frac{1}{5} \][/tex]
Breaking it down step by step:
- [tex]\(5 \times \frac{1}{5} = 1\)[/tex]
- [tex]\(10 \times \frac{3}{5} = 6\)[/tex]
- [tex]\(20 \times \frac{1}{5} = 4\)[/tex]
Adding these values together:
[tex]\[ E[X] = 1 + 6 + 4 = 11 \][/tex]
Thus, the expected value of the person's winning prize is [tex]\(\$11\)[/tex].
Therefore, the correct answer is:
(A) [tex]\(5 \times \left(\frac{1}{5}\right) + 10 \times \left(\frac{3}{5}\right) + 20 \times \left(\frac{1}{5}\right) = \$11\)[/tex]
