TITLE: A coin is flipped ten times. What is the probability that the first three are heads if an equal number of heads and tails are flipped?
QUESTION [14 upvotes]: I understand the question but I am not sure how to solve it. For example, if we flip HHHTTTTT then the next three must be heads because of the question. This however seems counterintuitive. I believe that there are $2^{10}$ possible strings, but I am unsure of how to count all possible strings that begin with HHH.

REPLY [3 votes]: There are $\frac{10!}{5!5!}$ ways of getting an equal number of heads and tails. The first three are fixed at heads so we need 2 more heads and 5 tails in the remaining seven tosses this can be done in $\frac{7!}{2!5!}$ ways. The probability is therefore
$$
{\frac{7!}{2!5!}\over\frac{10!}{5!5!}} = \frac{1}{12}
$$