Problem #5 (30 points) - Finding The Missing Ship

Suppose a distressed call is made to a Coast Guard Station from a pleasure

cruise that is sinking. The distress call is garbled so that all that can be

made out is that the sinking ship is near an island. Unfortunately, there are

two islands (call them #1 and #2) that could be the approximate location

of the sinking ship. Island #1 is ten miles to the south of the Coast Guard

Station and Island #2 is ten miles to the north of the Coast Guard Station.

Since time is of essence, the Coast Guard Station commander splits his fleet

of n boats and orders his boats to search water off of both islands. Because of

the time of day, the commander thinks that the probability that the sinking

ship is at Island #1 is p1 and the probability that the sinking ship is at

Island #2 is p2 = 1−p1. Assume that each search boat has probability ps of

finding the sinking ship if it is sent to the correct island, and that all search

boats act independently. How should the commander divide his fleet of

search boats in order to maximize the probability (P) of finding the sinking

ship? In other words, determine (in terms of p1, p2 and ps) the number of

boats (call it n1) that should be sent to Island #1 and the number of boats

(call it n2 = n−n1) that should be sent to Island#2 so that the probability

of finding the sinking ship is a maximum. Then determine your values of n1

and n2 for the numerical case in which p1 = 2/5, p2 = 3/5, ps = 1/10, and

n = 10, and also determine Pmax.