Sure, let's proceed step-by-step to calculate the expected value for a ticket in this raffle scenario.
1. Determine the Probability of Winning and Losing:
- Probability of Winning the Car: Since there is only one car and 5,000 tickets are sold, the probability of winning the car is given by:
[tex]\[
\text{Probability of Winning} = \frac{1}{5000} = 0.0002
\][/tex]
- Probability of Losing: If you do not win the car, you lose. Thus, the probability of losing is:
[tex]\[
\text{Probability of Losing} = 1 - \text{Probability of Winning} = 1 - 0.0002 = 0.9998
\][/tex]
2. Determine the Values Associated with Winning and Losing:
- Value if Win: If you win, you get the car worth [tex]$30,000. Thus, the value of winning can be stated as:
\[
\text{Value if Win} = 30,000
\]
- Value if Lose: If you lose, you are out the cost of the ticket, which is $[/tex]20. Thus, the value of losing is:
[tex]\[
\text{Value if Lose} = -20
\][/tex]
3. Expected Value Calculation:
- The expected value (EV) of a ticket can be calculated using the formula:
[tex]\[
EV = (\text{Value if Win} \times \text{Probability of Winning}) + (\text{Value if Lose} \times \text{Probability of Losing})
\][/tex]
Plug in the values:
[tex]\[
EV = (30,000 \times 0.0002) + (-20 \times 0.9998)
\][/tex]
Simplify the terms:
[tex]\[
EV = 6 - 19.996
\][/tex]
[tex]\[
EV = -13.996
\][/tex]
Hence, the expected value (EV) for a ticket in this raffle is:
[tex]\[
-13.996
\][/tex]
Therefore, among the given options, none of them directly represent the correct calculation or expected value [tex]$-13.996$[/tex].
The correct calculation should be:
[tex]\[
30,000 \left( \frac{1}{5000} \right) + (-20) \left( \frac{4999}{5000} \right) = E(X)
\][/tex]
[tex]\[
\left( 30,000 \times 0.0002 \right) + (-20 \times 0.9998) = E(X)
\][/tex]
This confirms that the expected value of purchasing a ticket in this raffle is approximately [tex]$-13.996$[/tex].
