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Coin Tossing Games

Age 14 to 16
Challenge Level Yellow starYellow star
Secondary curriculum
  • Problem
  • Student Solutions

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)?


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The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

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