Consecutive Tail-Tail or Tail-Head

Question

If A and B play a game with a fair coin such that A wins by getting consecutive TT for the first time, while B wins by getting a consecutive TH. Who has higher probability of winning the game?

I tried the following method but not sure:

For A to win,

TT, HTT, HHTT ... that is, 2^-k for each game length k

For B to win,

TH, HTH, HHTH ... that is, 2^-k for each game length k

So ultimately I am getting same probability of each to win.

@DonThousand Not true. for instance, HT has a probability of $75%$ to appear before TT (basically, as long as the first two throws aren't TT, HT will come first). — Arthur, Mar 19 '19 at 08:51
So, they toss the same coin? Or each is tossing her own coin, with A starting first? — Jimmy R., Mar 19 '19 at 09:00
@JimmyR "a game with a fair coin" seems to imply that there is a single coin, and whoever has their sequence appear first wins. — Arthur, Mar 19 '19 at 10:51
@SayangdiptoChakraborty He thought that just because they are two equally long sequences, they are equally likely to appear first. But that argument is flawed, as is evidenced by my example. — Arthur, Mar 19 '19 at 11:32
@JimmyR. It was my mistake in understanding the problem. You can remove your downvote. — Sayangdipto Chakraborty, Mar 19 '19 at 11:52

score 2 · Accepted Answer · answered Mar 19 '19 at 08:54

2

As the first T happens, none of them have won yet, and on the next throw after the first T, one of them will win. If that next throw is a T, then A wins, and if it's a H then B wins. That makes it the same probability for either of them to win.

answered Mar 19 '19 at 08:54

Arthur

199,419

are you sure because even I have tested with all possible combinations upto game length 5 but still can't generalise. – Sayangdipto Chakraborty Mar 19 '19 at 09:49
However these two answers [https://math.stackexchange.com/questions/534800/probability-that-a-need-more-coin-tosses-to-get-two-consecutive-heads-than-b?rq=1] confused me – Sayangdipto Chakraborty Mar 19 '19 at 09:52
And this one too [https://math.stackexchange.com/questions/1450884/fair-coin-tosses-until-until-two-consecutive-heads-or-two-consecutive-tails-appe?rq=1] – Sayangdipto Chakraborty Mar 19 '19 at 09:54
@SayangdiptoChakraborty The game will start with $k$ times $H$, then a $T$ (and both players hold their breath) and then a decisive throw that has equal chance to be in favor of $A$ and of $B$. Here $k$ is a nonnegative integer. – drhab Mar 19 '19 at 10:59
@drhab Got it, thank you! – Sayangdipto Chakraborty Mar 19 '19 at 11:53

Consecutive Tail-Tail or Tail-Head

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