Two persons A and B have respectively ($n+1$) and $n$ coins which they toss simultaneously. The probability that A will have more number of heads is

For every outcome where A gets k heads, B can get fewer, equal, or more heads than A. By the symmetry of tossing coins, the probability that B gets more heads than A equals the probability that B gets fewer heads than A. If the probabilities of B having more, fewer, and equal heads are p, p, and q respectively, then 2p + q = 1. When A has n+1 coins and B has n coins, the probability of a tie is q. The probability that A gets exactly k heads is the sum of A getting k heads and B getting fewer than k heads. The complementary probability is A getting k heads and B getting k or more heads, which evaluates to 1/2. Therefore, the probability is 1/2.

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