On Friday we posed the following 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 (£2^{n}) 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?

Well, consider a simpler game where a head pays out £10 and a tail pays out £4. The probability of winning £10 is 50%, as is the probability of winning £4. So the average winnings are £(10+4)/2 = £14/2 = £7. If you bid anything less than £7 you will be a winner in the long run.

However, in this game the probability of winning £2 is 50%, of winning £4 it is 25%, and so on . . . to infinity. And if you sum 2/2 + 4/4 + 8/8 to infinity, you get infinity. In other words, the expected winnings (in the long term) are infinite.

So logically a gambler should pay any amount to enter the game as, if the game is played for long enough, they will always return a profit. But no rational gambler would do this, in spite of the calculated returns.

And this is the St. Petersburg Paradox!