Assume that a dart is randomly thrown at the dartboard shown here and strikes the board every time. The payoffs are listed on the board. How much should you be willing to pay for the opportunity to play this game? (Give your answer to the nearest cent.)
$32 -$11
$16 $4 $2
$8
$ 1
cent
To determine how much you should be willing to pay for the opportunity to play this game, you need to calculate the expected value.
The expected value is calculated by multiplying the payoffs by their respective probabilities and summing them up.
Let's assign probabilities to each outcome based on the given information:
- Probability of hitting $32: 1/12
- Probability of hitting -$11: 1/6
- Probability of hitting $16: 1/3
- Probability of hitting $4: 1/6
- Probability of hitting $2: 1/18
- Probability of hitting $8: 1/18
- Probability of hitting $1: 1/12
Now, calculate the expected value:
Expected value = (Probability of Outcome 1 * Payoff of Outcome 1) + (Probability of Outcome 2 * Payoff of Outcome 2) + ...
Expected value = (1/12 * $32) + (1/6 * -$11) + (1/3 * $16) + (1/6 * $4) + (1/18 * $2) + (1/18 * $8) + (1/12 * $1)
Expected value = $2.666667
Therefore, you should be willing to pay approximately $2.67 for the opportunity to play this game.
To determine how much you should be willing to pay for the opportunity to play this game, you need to calculate the expected value. The expected value is the average value of each possible outcome, weighted by the probability of each outcome.
In this case, we can calculate the expected value by multiplying each payoff by its probability and summing them up. Let's break down the probabilities:
- The probability of hitting the $32 payoff: It is not mentioned, so let's assume it is equidistant and has a probability of 1/6.
- The probability of hitting the -$11 payoff: It is not mentioned, so let's assume it is equidistant and has a probability of 1/6.
- The probability of hitting the $16 payoff: It is not mentioned, so let's assume it is equidistant and has a probability of 1/6.
- The probability of hitting the $4 payoff: It is not mentioned, so let's assume it is equidistant and has a probability of 1/6.
- The probability of hitting the $2 payoff: It is not mentioned, so let's assume it is equidistant and has a probability of 1/6.
- The probability of hitting the $8 payoff: It is not mentioned, so let's assume it is equidistant and has a probability of 1/6.
- The probability of hitting the $1 payoff: It is not mentioned, so let's assume it is equidistant and has a probability of 1/6.
Now, let's calculate the expected value:
Expected value = (32 * 1/6) + (-11 * 1/6) + (16 * 1/6) + (4 * 1/6) + (2 * 1/6) + (8 * 1/6) + (1 * 1/6)
Expected value = (32/6) + (-11/6) + (16/6) + (4/6) + (2/6) + (8/6) + (1/6)
Expected value = 61/6 ≈ $10.17
Therefore, you should be willing to pay around $10.17 for the opportunity to play this game.