Let's break down the problem step by step to calculate the expected value of buying a ticket in this raffle.
1. Define the Variables:
- The winning prize is \[tex]$100.
- The cost of each ticket is \$[/tex]5.
- The total number of tickets sold is 500.
2. Calculate the Probability of Winning:
The probability of winning (denoted as [tex]\(P(\text{win})\)[/tex]) is given by:
[tex]\[
P(\text{win}) = \frac{1}{\text{total number of tickets}} = \frac{1}{500}
\][/tex]
3. Calculate the Probability of Not Winning:
The probability of not winning (denoted as [tex]\(P(\text{not win})\)[/tex]) is:
[tex]\[
P(\text{not win}) = 1 - P(\text{win}) = 1 - \frac{1}{500} = \frac{499}{500}
\][/tex]
4. Calculate the Expected Value of Winning:
The expected value of winning (denoted as [tex]\(E(\text{win})\)[/tex]) is the prize multiplied by the probability of winning:
[tex]\[
E(\text{win}) = 100 \left(\frac{1}{500}\right) = 0.2
\][/tex]
5. Calculate the Expected Value of Not Winning:
The expected value of not winning (denoted as [tex]\(E(\text{not win})\)[/tex]) involves the cost of the ticket (since you lose this much when you don't win) multiplied by the probability of not winning:
[tex]\[
E(\text{not win}) = -5 \left(\frac{499}{500}\right) = -4.99
\][/tex]
6. Sum These Expected Values:
The total expected value of buying a ticket (denoted as [tex]\(E(X)\)[/tex]) is the sum of the expected values of winning and not winning:
[tex]\[
E(X) = E(\text{win}) + E(\text{not win}) = 0.2 + (-4.99) = -4.79
\][/tex]
Putting it all together, the correct equation that calculates the expected value should be:
[tex]\[
100\left(\frac{1}{500}\right)+(-5)\left(\frac{499}{500}\right)=E(X)
\][/tex]
So, the correct answer is indeed:
[tex]\[
100\left(\frac{1}{500}\right)+(-5)\left(\frac{499}{500}\right)=E(X)
\][/tex]
