A carnival gave offers a 100 cash prize for anyone who can b

A carnival gave offers a $100 cash prize for anyone who can break a balloon by throwing a dart at it. It costs $5 to play, and you\'re willing to spend up to $20 trying to win. You estimate that you have about 10% chance of hitting the balloon on any throw.

1. Create a probability model for this carnival game. (hint: give the model in the form of a probability distribution table)

2. Find the expected number of darts you\'ll throw.

3. Find your expected winnings.

Solution

Let X shows the number of throws until balloon break. Since you are willing to spend up to $20 trying to win so you are willing to try maximum four throw. That is X can take values 1, 2, 3, and 4. Since the probability that you hit the balloon on any throw is 10% so probability of success is p=0.10. On the fourth trial, it may be possible that you will fail. So pdf of X is

2.

Following table shows the calcualtions for expected number of throws:

So expected number of throws is 3.439.

3.

Let Y shows the amount you win or gain. Following table shows the calcualtions:

Following table shows the calcualtion for expected values:

So expected winning are :

E(Y)=$17.195