About Last Month
So as I mentioned last time, both of these have relatively simple solution. For the first problem, calculating probabilities is a pretty futile exercise. You can try it for a small value of n and do it, but if n is--say--1,000,000, things will get out of hand in a hurry. The trick to this problem is just to note that if you throw 2n+1 coins, you will either throw n+1 heads or n+1 tails, but not both. Since the probability of each of these events is the same by symmetry, the probability that either player wins must be exactly 1/2.