Explore a world of knowledge and get your questions answered on IDNLearn.com. Get the information you need from our experts, who provide reliable and detailed answers to all your questions.
Sagot :
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].
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].
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.