Tossing two coins 100 times each, what is the probability for same number of heads?

Question

I know it's a binomial distribution, but haven't been able to figure out how to do it. Well there'll be two identical distributions, and then I'm not sure how to proceed. I could multiply them, couldn't I?

Answer 1

There are $2^{200}$ possible outcomes of the coin flips. Of these, $\binom{100}n^2$ of them have both coins showing heads $n$ times each, so the number of outcomes we want to count is $$\sum_{n=0}^{100}\binom{100}n^2$$ which is known to be equal to the central binomial coefficient (OEIS A984) $$\binom{200}{100}$$ The probability we want is thus $$\frac{\binom{200}{100}}{2^{200}}$$