You are playing Monopoly with a friend. She owns and has developed Park Place an

Question

You are playing Monopoly with a friend. She owns and has developed Park Place and Boardwalk with hotels. If you roll an 8, you will land on Park Place and owe her $1500.

You will be able to pay, but you will not have money left afterward. If you roll a 9, you will land on Luxury Tax and have to pay $75.

If you roll a 10, you will land on Boardwalk and owe her $2000, which is more than you can pay—and lose the game. If you roll a 7 or less, you will not have to pay any money to anyone.

Your friend offers you insurance. Pay her $500 before you roll and even if you land on Boardwalk or Park Place, you will not have to pay any additional money. However, if you roll a 7, you will have to pay her $1000. Use probabilities to find expected values. Then compare the amount of money you should expect to pay out on average, under her insurance and by chance. Is her deal fair?

Explanation / Answer

First recognize that the probability of rolling a 7 with two die is 6/36, an 8 is 5/36, etc.

Suppose you ignore the insurance, and risk losing the game. Then, the expected payout is:
(5/36)*-1500 + (4/36)*-75 + (3/36)*-2000 + (24/36)*0 = -383.33.

Under the insurance (assuming you still have to pay luxury tax if you land on it):
(5/36)*0 + (4/36)*-75 + (3/36)*0 + (18/36)*0 + (6/36)*-1000 -500 = -675
(no matter what, you lose 500 if you decide to buy insurance)

Hence, her deal is not fair. You can however make an argument that because there's a chance you will lose the game if you don't buy insurance, the deal may be the wise thing to do. But in terms of expected payouts, not buying insurance is better.

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