2007 Alabama ARML TST Problems/Problem 13

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Problem

Before he gets out of bed every morning, Calvin the Compulsive plays a game with a fair coin. He flips it until either he flips four consecutive heads or he flips six consecutive tails, then he immediately gets out of bed and brushes his teeth. If his last flip is a head, he eats two melons for breakfast. Otherwise, he eats just one. Find the probability that Calvin ate two melons for breakfast this morning.

Solution

We will say that Calvin wins the game iff he eats two melons.

Consider these two situations:

• : The last run of equal throws contains exactly one head.
• : The last run of equal throws contains exactly one tail.

That is, situation occurs either after the very first throw (if it was a head), or after a sequence of throws that ends with "... tail head".

Let be the probability that Calvin wins the game if he is now in situation , and similarly let be the probability of winning from .

We can now make the following observations:

When in the situation , we have probability of winning the game right away, by throwing three more heads in a row. With probability this does not happen, and we throw a tail. The first tail we throw takes us into the situation .

Similarly, from situation either we lose right away, which happens with probability , or we get into situation .

This gives us two equations for and :

This solves to and .

Now, from the initial state the first throw takes us either to situation or to situation , with equal probability. Thus the answer is .

See also