The coast guard station has received a distress call from the S.S. Minnow, sinking near one of two islands in opposite directions from the station. The call was interrupted by radio failure on the Minnow before the tourboat could specify which of the two islands it was near.
The station chief knows from experience with that tour operator that there is a 20% chance the boat is near island A and an 80% chance it's near island B. The station has 13 rescue boats, and, again from experience, it is known that each rescue boat has, independently of the other search boats, a 20% probability of finding a distressed boat if indeed a distressed boat is present, effecting a rescue.
How should the 13 boats be split between the two islands to maximize the probability that the people aboard the Minnow will be rescued? What is the probability that they will in fact be rescued if that optimal strategy is followed?
Part 2:
Suppose the coast guard station has 40 boats available but each one has only a 5% probability, independently, of finding a ship in trouble (given there is one in that location to be found). And further, there's only a 10% probability the boat is near island A, 90% of being near island B.
for part 1.
The likelihood the Minnow will _not_ be found is the prob it is at A and all "a" boats fail to find it + the likelihood it is at B and all (13-a) boats fail.
The likelihood of success is 1-p
1-p = 1- [0.2 * 0.8^a + 0.8 * 0.8^(13-a)]
This function maximizes at a = 3
Prob of rescue = 1-p = 0.8117006
For part 2.
1-p = 1- [0.1 * 0.95^a + 0.9 * 0.95^(40-a)]
This peaks at a =0: send all the ship to Isle B!
Prob of rescue = 1-p = 0.7843391...
I checked with a little simulation repeating the rescue 10^8 times
with each split of the fleet.
theory likelihood found = 0.784339130 a = 0
sim likelihood found = 0.784281671 a = 0
theory likelihood found = 0.783251703 a = 1
sim likelihood found = 0.783144891 a = 1
theory likelihood found = 0.781593919 a = 2
sim likelihood found = 0.781467795 a = 2
theory likelihood found = 0.779361367 a = 3
sim likelihood found = 0.779310286 a = 3
Edited on September 3, 2020, 1:40 am
Soln (the Minnow might be lost)
