Coin Tossing Games
Set by Dr Susan Pitts, University of Cambridge Statistics Laboratory,
for the Summer 1997 NRICH Maths Club Video-conference.

You and I play a game involving successive throws of a fair coin. Let H and T denote heads and tails respectively.

I pick HH. Suppose that you pick TH. The coin is thrown repeatedly until we see either two heads in a row or a tail followed by a head. In the first case I win; in the second case you win. What is the probability that you win?

What is the probability that you win if you choose HT? Or TT? What is the best choice you can make?

What should you choose if I choose TT?

What happens if I choose HT?

Assuming that you always make a choice that maximises your chance of winning, what should I choose to maximise the probability that I win?

Now suppose that we look at triples instead of pairs. What is the probability that you win if I choose HHH and you choose THH?

I have eight possible choices. For each one, can you find a triple that gives you a better than even chance of winning (i.e. a triple that makes your probability of winning more than 1/2)?