A batch of 100 motors has been manufactured and Andy and Bob

A batch of 100 motors has been manufactured, and Andy and Bob are applying a standard stress test to them. Andy is assigned 40 motors to test. and Bob is assigned the remaining 60 motors to test. Each motor has probability p of falling: assume the motors and tests are independent. Let X be the number of failures among Andy?s motors, Y be the number of failures among Bob?s motors, and let T = X + Y. Calculate the conditional probability P(X = x|T = t)

Solution

Note that, when Andy has X failures and together they have T failures, then Bob has T - x failures.

Thus, P(X=x|T=t) = P(X=x, y = T-X)/P(T=t) =

C(40,x)p^x(1-p)^(40-x)*C(60,T-x)p^(T-x)(1-p)^(60-(T-x))/

C(100,T)p^T(1-p)^(100-T) =

C(40,x)C(60,T-x) p^T(1-p)^(100-T)/C(100,T)p^T(1-p)^(100-T) =

C(40,x)C(60,T-x)/C(100,T)

The terms involving p cancel out.

If we wish we may expand this as

40!/(x!(40-x)!)60!/(T-x!(60-T+x)!)/(100!/T!(100-T!)) =

40!60!T!(100-T)!/ (100!x!(40-x)!T-x!(60-T+x)!)