Pregunta
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For a fundraiser, there is a raffle with 1,000 tickets. One ticket will win a prize,
even tickets will win a prize, nine tickets will win a prize, and the rest will win
othing. If you have a ticket, what is the expected payoff?

Ask by Willis Stephens. in the United States
Jan 23,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

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The expected payoff for a ticket is $116.10.

Solución

To find the expected payoff for a ticket in the raffle, we need to calculate the expected value based on the probabilities of winning each prize and the corresponding amounts.

Known Conditions:

  1. Total tickets:
  2. Prizes:
    • 1 ticket wins
    • 500 tickets (even tickets) win
    • 9 tickets win
    • The rest win nothing.

Step-by-Step Calculation:

  1. Calculate the probabilities of winning each prize:
    • Probability of winning :
    • Probability of winning :
    • Probability of winning :
    • Probability of winning nothing:
  2. Calculate the expected value (EV):
    The expected value is calculated as follows:
    Substituting the probabilities and prizes:
  3. Perform the calculations:
    • For :
    • For :
    • For :
    • For nothing:
  4. Sum the expected values:

Final Answer:

The expected payoff for a ticket in the raffle is .

Respondido por UpStudy AI y revisado por un tutor profesional

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Extra Insights

To determine the expected payoff for a single ticket, we first identify the prizes and their probabilities. There is 1 grand prize of , 500 even tickets (2, 4, 6, etc.) each winning , and 9 tickets winning . Out of 1,000 tickets, that means 490 tickets win nothing.
Now, we calculate the expected payoff:
  • Grand Prize: ticket wins with a probability of .
  • Even Prize: tickets win with a probability of .
  • Minor Prizes: tickets win with a probability of .
  • No Prize: tickets win with a probability of .
Now, we calculate the contribution from each possible outcome:
  1. Grand Prize:
  2. Even Prize:
  3. Minor Prizes:
  4. No Prize:
Adding these together gives the expected payoff:
Thus, the expected payoff for one ticket is:
Expected Payoff = $116.1

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