Question 134
Three brothers Abel, Banach, and Gauss each have portable music players that can share music with each other. Initially, Abel has $9$ songs, Banach has $6$ songs, and Gauss has $3$ songs, and none of these songs are the same. One day, Abel flips a coin to randomly choose one of his brothers and he adds all of that brother's songs to his collection. The next day, Banach flips a coin to randomly choose one of his brothers and he adds all of that brother's collection of songs to his collection. Finally, each brother randomly plays a song from his collection with each song in his collection being equally likely to be chosen. What is the probability that they all play the same song?