If a raffle has a winning prize of [tex]$\$[/tex]100[tex]$ and each ticket costs $[/tex]\[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 :

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]