GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
220 views

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_____.

asked in Probability by Junior (685 points)   | 220 views

1 Answer

+7 votes
Best answer

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

answered by Veteran (65k points)  
selected by

Related questions

0 votes
0 answers
1
asked ago in Probability by Vignesh Sekar (369 points)   | 20 views
Top Users Feb 2017
  1. Arjun

    5386 Points

  2. Bikram

    4230 Points

  3. Habibkhan

    3952 Points

  4. Aboveallplayer

    3086 Points

  5. Debashish Deka

    2564 Points

  6. sriv_shubham

    2318 Points

  7. Smriti012

    2236 Points

  8. Arnabi

    2008 Points

  9. mcjoshi

    1696 Points

  10. sh!va

    1684 Points

Monthly Topper: Rs. 500 gift card

20,863 questions
26,021 answers
59,689 comments
22,131 users