AIME I 2010 #4

[stupidityismygam]

Jackie and Phil have two fair coins and a third coin that comes up heads with probability [unparseable or potentially dangerous latex formula]. Jackie flips the three coins, and then Phil flips the three coins. Let [unparseable or potentially dangerous latex formula] be the probability that Jackie gets the same number of heads as Phil, where [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are relatively prime positive integers. Find [unparseable or potentially dangerous latex formula].

[88bobcat]

[stupidityismygam]

The situation follows the generating function of [unparseable or potentially dangerous latex formula]. Expanding, we get [unparseable or potentially dangerous latex formula].

The desired probability is simply the sum of the squares of the coefficients (in the generating function the coefficients represent probabilities), so the answer is [unparseable or potentially dangerous latex formula]