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The treasure is hidden on one of n islands, labeled from 1 to n. The probability that the treasure is on island i is pa,

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The treasure is hidden on one of n islands, labeled from 1 to n. The probability that the treasure is on island i is pa, where pit..+ Pn =1. A pirate looks for the treasure. If the treasure is on island i and the pirate searches the island, the pirate will find the treasure with probability a;. Suppose that the pirate searched island i and did not find the treasure. Show that given this new information, it is reasonable that the pirate updates the probability that the treasure is on island j is equal to Pi (1 - a;)Pi 1 - aiPi for j # i and 1 - a;Pi for j =

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