To solve for the expected value in this problem, let’s break down the steps involved:
1. calculate the probability of winning:
- The raffle has a total of 500 tickets.
- The probability of winning is given by [tex]\(\frac{1}{500}\)[/tex].
2. calculate the probability of losing:
- Only one ticket wins, so the probability of losing is given by [tex]\(\frac{499}{500}\)[/tex].
3. calculate the winnings and losses:
- If you win, you get \[tex]$100.
- The cost of each ticket is \$[/tex]5, so if you lose (which is most likely), you lose the \$5 spent on the ticket.
4. calculate the expected value:
- The expected value [tex]\(E(X)\)[/tex] is the sum of all possible outcomes, weighted by their probabilities.
- The expected value calculation formula in this context is:
[tex]\[
E(X) = (\text{Prize} \times \text{Probability of Winning}) + (\text{Loss} \times \text{Probability of Losing})
\][/tex]
Substituting in the numbers:
[tex]\[
E(X) = 100 \left(\frac{1}{500}\right) + (-5) \left(\frac{499}{500}\right)
\][/tex]
After summing these products:
[tex]\[
E(X) = 0.2 - 4.99 = -4.79
\][/tex]
Therefore, the correct equation that calculates the expected value [tex]\(E(X)\)[/tex] is:
[tex]\[
100 \left(\frac{1}{500}\right) + (-5) \left(\frac{499}{500}\right) = E(X).
\][/tex]
