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If a raffle has a winning prize of [tex]\(\$100\)[/tex] and each ticket costs [tex]\(\$5\)[/tex] with a total of 500 tickets sold, which equation would calculate the expected value?

A. [tex]\(100\left(\frac{1}{500}\right) + (-5)\left(\frac{499}{500}\right) = E(X)\)[/tex]

B. [tex]\(95\left(\frac{1}{500}\right) + (-5)\left(\frac{499}{500}\right) - E(X)\)[/tex]

C. [tex]\((100-5)\left(\frac{1}{500}\right) = E(X)\)[/tex]

D. [tex]\(100\left(\frac{1}{500}\right) = E(X)\)[/tex]



Answer :

GinnyAnswer
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]

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