A Peculiar Game
You and your best friend decide to play the following game:
Player 1 and Player 2 both start with $100. One round of this games consists of the following:
Both players choose a number randomly and independently from 1 to 5 inclusive. If both players choose the same number, then Player 1 gives $10 to Player 2. Otherwise, Player 2 gives $10 to Player 1.
You decide to be Player 1, and your best friend is Player 2. What is expected amount of money (in dollars) you will be left with after playing 10 rounds of this game?
The answer is 160.
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1 solution
The probability that the two players choose the same number is 5 1 and the probability that the two players choose a different number is 5 4 . Hence there is 5 1 chance that Player 1 has to give $ 10, and there is 5 4 chance that Player 1 gets $ 10 in each round.
The expected amount of money Player 1 gets in each round is
E [ X ] = ∑ x i P i = 5 1 × ( − 1 0 ) + 5 4 × ( + 1 0 ) = + 6
Player 1 is expected to get +$6 in each round. Thus after 10 rounds, Player 1 is expected to get 1 0 × 6 = + $ 6 0 .
Since Player 1 started out with $100, Player 1 is expected to be left with $ 1 6 0 at the end of 10 rounds. □