(Adapted from the minigame "Hide and Go Boom" from Nintendo's "Mario Party 4")
In this game, 1 player competes against 3 other players. Players 2, 3, and 4 independently select one of four slots (labeled A, B, X, and Y) in which to hide. More than one player can hide in the same slot and none of these three "hiders" can co-ordinate their actions. Once the three are hidden, player 1 gets to select 3 of the 4 slots in which to "find" the other three players. If all three hidden players were located in the three slots chosen by player 1, then player 1 wins. If any of the three hidden players are located in the slot not selected by player 1, then the group of players 2, 3, and 4 win.
(a) What is the probability that player 1 wins?
(b) Suppose the three-player team was allowed to coordinate their hiding efforts and they decided to all hide in the same slot. What then would be the probability that player 1 wins?
(c) Now, suppose the three-player team was allowed to coordinate their hiding efforts and they decided to all hide in different slots. Now what is the probability that player 1 wins?