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Coin Tossing Games

You and I play a game involving successive throws of a fair coin. Suppose I pick HH and you pick TH. The coin is thrown repeatedly until we see either two heads in a row (I win) or a tail followed by a head (you win). What is the probability that you win?

Bet You a Million

Age 14 to 16
Challenge Level

Here's how this bet works:

It involves nothing more than just tossing an ordinary coin until the first Head appears.
If it's a Head straight-off the prize is £2 and the game's over.
If instead it's a Tail the coin is tossed again.

The prize is now £4 for a Head if it shows, and the game will then be over.
But if it's a Tail, as before, the game continues.

The prize paid for the Head that finishes the game doubles each time a Tail turns up instead.

Not a bad game - you could get a very big prize.
On the other hand you may not get to the big prizes very often.

How many Tails would need to come up, before the winning Head, to win a million pounds or more ?
And how often would you expect to see that ?

So now for the real question :

How much would you be prepared to pay to play this game ?

Would £10 be too much ?

There'll be quite a lot of £2 and £4 wins but £16 and upwards would also come along from time to time.

You may assume for this question that you can magically avoid Gambler's Ruin

Note - Gambler's Ruin is when the person betting runs out of money before a big payout even though the odds were otherwise in their favour. Imagine playing a National Lottery, at £1 a time, continuously with ten million other people where there was only one prize each time but the prize was a million million pounds. How many people would have run out of money before it was their turn to win ?

No matter how big the prize or how easy it looks to win, it isn't smart to bet if we can't stand the loss.
However, lots of things are not certain and we often need to make decisions in the face of that uncertainty.
Probability is how mathematicians quantify uncertainty and puzzles and games can be an excellent way to explore this.