Make the House Win
In this problem , a casino wanted to introduce the following game and charge for every player that plays:
- The player flips a fair coin until he or she flips heads.
- The starting prize is (if the player flips heads on the first flip).
- The prize doubles with each flip after the first.
For example, if somebody flipped tails-tails-tails-heads, then his or her prize would be ( doubled three times).
However, the casino realized that the house will eventually lose with these rules when some lucky person flips an unusually high number of tails in a row, so the casino owner wants to revise the first rule to:
- The player flips a fair coin until he or she flips heads or until he or she flips the coin times .
Find the maximum value of so that the house can still expect to make a profit.
The answer is 18.
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1 solution
Let E ( x ) be the expected payout in relation to the maximum number of coin flips n . We need to find the value of n , such that E ( n ) < $ 2 0 and E ( n + 1 ) ≥ $ 2 0 .
Let k ≤ n . The probability of throwing k − 1 times tails and then throwing heads is ( 2 k ) − 1 , in which case we get $ 2 k . The probability of throwing n times tails is ( 2 n ) − 1 , in which case we get $ 2 n . Thus, E ( x ) is equal to E ( x ) = i = 1 ∑ x ( $ 2 i 2 i 1 ) + $ 2 x 2 x 1 = $ ( x + 1 )
Thus, E ( 1 8 ) = $ 1 9 < $ 2 0 and E ( 1 9 ) = $ 2 0 , which yields a solution of n = 1 8 .