More heads than tails in ten flips

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A fair coin is flipped ten times. What is the probability that there are strictly more heads than tails? Explain why the answer is less than one half, and give the answer for eleven flips as well.

Accepted Answer

193/512, about 37.7 per cent. With eleven flips it is exactly one half. The tie. Ten flips can produce exactly five heads and five tails, a tie. The number of ways is 10 choose 5 = 252 out of 2 to the 10 = 1,024 possible sequences, so P(tie) = 252/1024, about 24.6 per cent. Symmetry. "More heads" and "more tails" are mirror images, so they are equally likely, and together they make up everything that is not a tie. So P(more heads) = (1 minus 252/1024) / 2 = (772/1024) / 2 = 386/1024 = 193/512, about 0.377. Eleven flips. With an odd number of flips a tie is impossible, so "more heads" and "mor…

More heads than tails in ten flips

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