A carnival game offers a $120 cash prize for anyone who can break a balloon by throwing a dart at it.It costs $9 to play and you're willing to spend up to $36 trying to win.You estimate that you have a 10% chance of hitting the balloon on any throw.Find the expected amount you will win.Assume that throws are independent of each other.
A)$13.01
B)$10.32
C)-$14.76
D)-$11.16
E)$19.32

Question

Quizplus · Accepted Answer

Let X be the random variable representing the amount won.

There are two possible outcomes: 
- Winning the game and getting $120, with probability 0.1. 
- Not winning the game and losing $9, with probability 0.9.

Therefore, the expected value of X can be calculated as:

E(X) = 0.1($120) + 0.9(-$9) 
E(X) = $12 - $8.1 
E(X) = $3.90

Since you are willing to spend up to $36 trying to win, you can play the game up to 4 times (since 4 x $9 = $36). Therefore, the expected amount you will win by playing the game 4 times is:

4 x E(X) = 4 x $3.90 = $15.60

However, since you are willing to spend only up to $36 trying to win, you will subtract the amount spent on playing:

$15.60 - $36 = -$20.40

This means that, on average, you will lose $20.40 by playing the game 4 times.

Therefore, the best choice is not to play the game at all, since the expected value is negative.

A Carnival Game Offers a $120 Cash Prize for Anyone