Some thoughts on a coin-tossing problem

Extract

“A fair coin is tossed successively until either two heads occur in a row or three tails occur in a row. What is the probability that the sequence ends with two heads?”

Thus the problem was stated on the Cambridge Scholarship paper. My problem on being asked for help was to produce a solution using only sixth form methods, and hopefully a solution which would prove interesting and instructive. Certainly the problem had captured the imagination of my questioners, although the method for its solution was proving to be elusive.