GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
535 views
Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$. If $P(.)$ denotes the probability of the event, the maximum value of $P(A)P(B)$ is_____.
asked in Probability by Veteran (77.2k points)   | 535 views

3 Answers

+8 votes
Best answer
1/2 * 1/2 =1/4

P(A) + P(B) = 1, since both are mutually exclusive and A ∪ B = S.
When sum is a constant, product of two numbers becomes maximum when they are equal. So, P(A) = P(B) = 1/2
answered by Junior (649 points)  
selected by
+1 vote

Suppose E:Even, O:Odd ,S:Sample Space of Natural Numbers

Example of such an mutually exclusive event is P(E)=P(O)=1/2

P(E)+P(O)=S(All Natural Numbers)=1

So Ans is 

P(E)*P(O)= $\frac{1}{2}$* $\frac{1}{2}$=$\frac{1}{4}$

answered by Veteran (16.8k points)  
0 votes
Sample Space(S) - A set of all possible outcomes/events of a random experiment. Mutually Exclusive Events - Those events which can't occur simultaneously.   P(A)+P(B)+P(A∩B)=1   Since the events are mutually exclusive, P(A∩B)=0. Therefore, P(A)+P(B)=1   Now, we now that AM >= GM So, (P(A)+P(B))/2 >= sqrt(P(A)*P(B))   P(A)*P(B) <= 1/4
  Hence max(P(A)*P(B)) = 1/4.
  We can think of this problem as flipping a coin, it has two mutually exclusive events ( head and tail , as both can't occur simultaneously). And sample space S = { head, tail }   Now, lets say event A and B are getting a "head" and "tail" respectively. Hence, S = A U B.   Therefore, P(A) = 1/2 and P(B) = 1/2.   And, P(A).P(B) = 1 /4 = 0.25.   Hence option B is the correct choice.
answered by Loyal (3.2k points)  


Top Users May 2017
  1. akash.dinkar12

    3292 Points

  2. pawan kumarln

    1652 Points

  3. sh!va

    1650 Points

  4. Arjun

    1424 Points

  5. Bikram

    1372 Points

  6. Devshree Dubey

    1272 Points

  7. Debashish Deka

    1142 Points

  8. Angkit

    1044 Points

  9. LeenSharma

    904 Points

  10. srestha

    718 Points

Monthly Topper: Rs. 500 gift card
Top Users 2017 May 22 - 28
  1. Bikram

    458 Points

  2. Arnab Bhadra

    402 Points

  3. pawan kumarln

    278 Points

  4. Ahwan

    236 Points

  5. bharti

    194 Points


22,786 questions
29,121 answers
65,184 comments
27,661 users