So two people are flipping a fair coin $n$ times each. What's the probability that they both flip the same number of heads?
My current approach was to use a binomial, and sum up the cases when X (no of heads) = 0,1,...,n But was wondering if there was a better approach, i.e from sum of variables or a generating function for each pair of flips?
Call these people Alice and Bob.
The probability that Alice tosses $k$ heads and Bob tosses $k$ heads is the same as the probability that Alice tosses $k$ heads and Bob tosses $k$ tails, that is the same as the probability that Alice tosses $k$ heads and Bob tosses $n-k$ heads.
Adding up over all $k$, he probability that Alice and Bob toss the same number of heads is the same as the probability that Alice and Bob toss $n$ heads between them.
Use combinational formula $C_n^k = C_n^{n-k}$ to reduce computation by half. The probability is $$p = \frac{\sum_{i=1}^{\frac{n}{2}}{(C_n^i)}^2}{2^{2n}} = \frac{C_{2n}^n}{2^{2n}}.$$
