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This problem trains for: AMC-10, AMC-12.

A mission to Mars is being organized by NASA. Scientists at NASA have no idea how to solve the following planning problem, therefore you have to help them!

Ships of type A require a crew of 16 people and have a success probability of 96%. Such ships can transport 3 units of cargo per trip.

Ships of type B require a crew of 19 people and have a success probability of 95%. They are able to transport 4 units of cargo per trip.

In order to minimize the number of crew casualties, NASA should send:

Denote the efficiency of a ship with:

equation

where p is the probability of arriving to destination, n is the number of crew that perishes in case of accident, and c is the number of units of cargo transported per trip.

The expected amount of cargo transported by a ship is:

equation

The expected number of crew that perishes if a ship is lost is:

equation

The efficiency of the transport is given by the amount of cargo per casualty:

equation

and has to be maximized.

For a ship of type A, the efficiency is:

equation