# Gambling at St. Petersburg Casino

Discrete Mathematics Level 3

A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 2 dollars and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if a tail appears on the first toss, 4 dollars if a head appears on the first toss and a tail on the second, 8 dollars if a head appears on the first two tosses and a tail on the third, 16 dollars if a head appears on the first three tosses and a tail on the fourth, and so on. In short, the player wins $$2^{k}$$ dollars, where $$k$$ equals number of tosses ($$k$$ must be a whole number and greater than zero). In dollars, what would be a fair price to pay the casino for entering the game?

Note: This problem is not original. Credits to: Nicolaus Bernoulli (1687-1759).

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