You are playing a card game with two people, East and West. There is (i) a 25% p

Question

You are playing a card game with two people, East and West. There is (i) a 25% probability that East has the king of spades and West has the queen of spades, (ii) a 25% probability that East has the queen of spades and West has the king of spades, (iii) a 25% probability that East has both the king and the queen of spades, and (iv) a 25% probability that West has both the king and the queen of spades. If you ask West if she has the king or the queen, (i) if she has neither she will tell you she has neither, (ii) if she has one but not the other she will show you the one she has, and (iii) if she has both she will show you the king with 50% probability and she will show the queen with 50% probability. You ask West if she has the king or queen of spades and she shows you the queen. What is the probability she also has the king?

Explanation / Answer

The probability that West has the queen of spades is 100% so this probability can be ignored. We also know that there is a 100% chance that the king and queen of spades are in one of these two hands. Even though the original probability of West having both the king and the queen was 25%, each of these two cards had a 50% chance of being in either hand regardless of the combination statistics (the 25% statistics come from multiplying .5 * .5). Now that the queen has been exposed, the probability of the king being in the East hand versus the West hand is 50:50.

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