# Coin flipping problem with sequences of results

Two players (A and B) are playing a game. Player A randomly chooses a sequence of three possible coin flips (eg HTH, TTH, etc) from the possible 8 and then player B replies with his own (non-random) choice. Then they flip a coin until one of the two sequences appears. What is the probability of player B winning?

We can immediately tell that certain sequences are advantageous to player B, for example if player A chooses HHH, then the sequence THH is an automatic win unless the first three tosses come out Heads. The same for TTT. Are all the others fair for both?

• Each sequence is equally likely to appear. This is not about guessing the number of heads or tails, right? In your example, why do you says that $THH$ is an automatic win? If the first toss is head, player $B$ does not win. Obviously, the win probability is not greater than $50$%. – Vasya Mar 12 at 16:40
• @Vasya the idea is you keep flipping until one or the other sequence appears, so with THH vs. HHH, THH will win 7/8 of the time, only losing when the first 3 flips are HHH. – Ned Mar 12 at 16:50
• Spoiler alert: See Penney's game. The table indicates which sequence is least bad for A, and the resulting winning probability for B (expressed as odds). – Brian Tung Mar 12 at 17:32
• Penney's Game is about choosing one sequence in response to another sequence. This question is about choosing 3 sequences in response to 3 sequences /and finding the probability of winning. – user558317 Mar 13 at 1:57
• Thank you all for your replies, I did not know this game had a name (Penney's game). I could see why some of the sequences win 7/8 of the time, but not why others were better (for example HHT being better than HTH). Knowing the name of the game I can research it, much appreciated! – Nikos Vlaseros Mar 15 at 16:18