Robin and Williams are playing a game. An unbiased coin is tossed repeatedly. Robin wins as soon as the sequence of tosses HHT appears. Williams wins as soon as the sequence of tosses HTH appears. The game ends when one of them wins. What are the probabilities of winning for each player?
Solution: (Robin) HHT – 2/3 (Williams) HTH – 1/3
Let the probability of Robin winning be p. The probability of Williams winning is (1-p). If the first toss is tails, it is as good as the game has not started, hence the probability of Robin winning is p after the first tail.
p = (1/2)*p + ….
Let the first toss be heads. If the second toss is heads, then Robin definitely wins. Since HH has occurred, and at some point, tails will occur, so HHT will occur. Hence Robin wins with probability 1 for HH.
p = (1/2 )*p + (1/2)*((1/2)*1 + .....)
Let the second toss be tails. If the third toss is heads, Robin loses as HTH occurs. If the third toss is tails (HTT) – since two tails have occurred in a row, now it is as good as the game has started from the beginning, so the chances of Robin winning are back to p.
T HH HTH HTT
p = (1/2)*p + 1/2 ((1/2)*1 + 1/2 ((1/2)*0 + (1/2) * p))
p = (1/2)*p + (1/4)*1 + (1/8)*0 + (1/8)*p
Finally, solving this equation gives us p = 2/3.