A Coast Guard dispatcher receives an SOS from a ship that has run aground off th

Question

A Coast Guard dispatcher receives an SOS from a ship that has run aground off the shore of of a small island. Before the captain can relay her exact position, her radio goes dead. The dispatcher has 12 helicopter crews he can send out to conduct a search. He suspects the ship is somewhere either south in area I (with probability 0.2) or north in area II (with probability 0.8). Each of the rescue parties is equally competent and have probability 0.3 of locating the ship, GIVEN that it has run aground in the sector being searched. He sends 4 crews to area 1 and 8 crews to area II. Find the probability that the ship is found.

Explanation / Answer

n=12

p=0.2

q=0.8

r=0.3

m=4

Let B denotes the event that the ship is found.

P(At least one helicopter finds the ship|Ship is in area I) = 1-P(None helicopter finds the ship|Ship is in area I)

= 1 - (1 - r)m

= 1 - (1 - 0.3)4

= 0.7599

P(At least one helicopter finds the ship|Ship is in area II) = 1-P(None helicopter finds the ship|Ship is in area II)

= 1 - (1 - r)n-m

= 1 - (1 - 0.3)12-4

= 0.94235199

Therefore:

P(B) = p.[1 - (1 - r)m] + (1 - p).[1 - (1 - r)n-m]

        = 0.2(0.7599) + 0.8(0.94235199)

        = 0.905861592

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