A small boy is lost coming down Mount Washington. The leader of the search team

Question

A small boy is lost coming down Mount Washington. The leader of the search
team estimates that there is a probability p that he came down on the east
side and a probability 1 - p that he came down on the west side. He has n
people in his search team who will search independently and, if the boy is
on the side being searched, each member will find the boy with probability
u. Determine how he should divide the n people into two groups to search
the two sides of the mountain so that he will have the highest probability of
fnding the boy. How does this depend on u?

Explanation / Answer

Let x,y be the number of peoples in group finding the boy in the east side and west side repectively. => P=P(found the boy) = P(the boy came down to east)*x*u + P(the boy came down to west)*y*u = u*(px+(1-p)*y) = u*(p(n-y)+(1-p)y) = u*(np+(1-2p)y) If p=1/2 => P=unp = constant => the leader can separate in arbitrary way. p (1-2p)>0 => Pmax when (1-2p)*ymax => ymax=n or the leader should send all the peoples to west side p>1/2 => (1-2p) to maximize P => minimize y => y=0 or send all the peoples to east side

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