I have been reading the randomized algorithm book by Rajeev Motwani and Prabhakar Raghavan. In section 3.5 they have introduced principle of deferred decision which is a different probability space. The example they provided is a clock solitaire game. The game is as follows. Initially 52 cards are randomly grouped into 4 cards of 13 piles. The piles are labeled $1,2,3,...,10,Q,J,K,A$. The game starts by drawing a card from $K$ labeled pile. The next draw will be taken place from the pile with the face value of the drawn card. For an example suppose the drawn card is 7, then we go to the pile labeled $7$ and pick a card from this pile. We continue in this fashion. The game ends when we reach an empty pile. And one wins if all the piles are empty when the game ends. It is easy to show that the last card drawn must have face value $K$.Now, to analyze the probability of winning, the author assumes a different probability space named Principle of deferred decision. The idea is to let the random choices unfold with the progress of the game, rather than fix the entire set of choices in advance. With this, they conclude that the probability of winning i.e., the probability that 52nd card being $K$ is 1/13. Can anyone explain why this is 1/13?
Clock solitaire game and principle of deferred decision
probability theory;randomized algorithms;probabilistic algorithms