probability

Question

Let S be a sample space and two mutually exclusive events Aand B be such that A∪B=A∪B=S. If P(.)denotes the probability of the event, the maximum value of P(A)P(B) is_____.

Answer 1

According to law of probability or set theory :

         P(A U B)  =  P(A)  +  P(B) - P(A ∩ B)

==>   1             =  x  + y    [ As A and B are mutually exclusive so P(A ∩ B)  =  0 and A U B = S so P(A U B) = 1 ]

==>   y             =  1 - x

So P(A).P(B)     =  x.y

                        =  x.(1 - x)

                        =  x - x²  = z(say)

Now for P(A).P(B) to be maximum ,

          dz / dz  =  0 

==>    1 - 2x  =  0

==>    x  =  1/2

Now taking 2nd derivative we get -2 which is < 0 and hence x = 1/2 is a point of maxima..

Therefore [P(A).P(B)]_max   =  (1/2)*(1/2)

                                      =  1 / 4