Gambling Game Puzzle

There is a popular new gambling game on offer at the casino, and it's very simple to play. However, this simple game leads to a very tricky puzzle.

As we said, the game is simple. The casino repeatedly tosses a fair coin until it lands on a tail. At that point the game ends and the casino pays out is as follows:

If the first toss is a tail they pay out £2.

If the first toss is a head and the second toss is a tail they pay out £4.

If the first two tosses are heads and the third toss is a tail they pay out £8.

And so on. Mathematically the prize can be expressed as 2 to the power n pounds (£2n) where n is the number of tosses needed to for the coin to land on a tail.

This seems to be a sure-fire winner, with a prize every time. And because of this there is a fee to play the game. Before each game the casino holds an auction and only the winner plays that game.

Clearly it's worth bidding something - £2 will guarantee your money back, with a very good chance of winning more. But is it worth bidding more? Indeed, how much is it worth bidding? What is the biggest bid you could make and statistically make money in the long run?

And that is the question - as usual the answer will be given at 9.00 on Monday. Please comment/tweet/post with your thoughts on the conundrum.

To help you think about this, the probability of a tail on the first toss is 50%, with a prize of £2. The probability of a head followed by a tail is 25%, with a prize of £4. And so on . . . forever!


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