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.)
$48 -$13
$24 $6 $3
$12
$
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 each outcome (payoff) by its probability of occurring, and then summing all the results.
Let's calculate the expected value for this game:
Payoffs:
$48 with probability 1/6
-$13 with probability 1/6
$24 with probability 1/6
$6 with probability 1/6
$3 with probability 1/6
$12 with probability 1/6
Expected value = (48 * 1/6) + (-13 * 1/6) + (24 * 1/6) + (6 * 1/6) + (3 * 1/6) + (12 * 1/6)
Expected value = 8 - 2.1667 + 4 + 1 + 0.5 + 2
Expected value = 13.3333 - 2.1667 + 4 + 1 + 0.5 + 2
Expected value = 19.6667
Therefore, the expected value for this game is $19.6667.
As a general rule, if you are risk-neutral, you would be willing to pay up to the expected value to play this game. Therefore, you should be willing to pay up to $19.67 (rounded to the nearest cent) 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 of the payoffs.
1. Multiply the value of each possible outcome by its corresponding probability.
- Probability of landing on $48: 1/9 (since there is only one target with $48 out of the 9 total targets)
- Probability of landing on -$13: 1/9
- Probability of landing on $24: 1/9
- Probability of landing on $6: 3/9 (since there are three targets with $6 out of the 9 total targets)
- Probability of landing on $3: 1/9
- Probability of landing on $12: 2/9 (since there are two targets with $12 out of the 9 total targets)
2. Sum up the products from step 1 to calculate the expected value.
- Expected value = ($48 * 1/9) + (-$13 * 1/9) + ($24 * 1/9) + ($6 * 3/9) + ($3 * 1/9) + ($12 * 2/9)
3. Calculate the final answer.
- Add up the products from step 2 and round to the nearest cent.
- The result will indicate how much you should be willing to pay for the opportunity to play this game.