The Puzzle of the Stranded Man

A man is stranded on an island. A benevolent genie presents three boxes, 23 white marbles, and 7 black marbles and instructs the man, "You may distribute the marbles into the boxes any way you see fit, but you must use all of the marbles. Once you finish, you will choose a box at random and then choose a marble from that box at random. If the marble is white, then I will help you escape from this place."

Assuming the man distributes the marbles in his best interest, what is the probability that he escapes the island?

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