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Question

If A and B are two events such that P left parenthesis A union B right parenthesis equals 5 over 6 comma P left parenthesis A intersection B right parenthesis equals 1 third and P left parenthesis A right parenthesis equals 2 over 3 then A and B are

  1. dependent events
  2. independent events
  3. mutually exclusive events
  4. mutually exclusive and independent events

Hint:

In this question we have to find what A and B events are if A and B are two events such that P left parenthesis A union B right parenthesis equals 5 over 6 comma P left parenthesis A intersection B right parenthesis equals 1 third and P left parenthesis A right parenthesis equals 2 over 3 .A and B are two events associated with the same random experiment, then A and B are known as independent events if P (A ∩ B) = P(B) .P(A), dependent events if  P(A∩B) = P(A).P(B/A) or P(A∩B) = P(B).P(A/B) and mutually exclusive if P (A ∩ B) =0.

The correct answer is: independent events


    As P left parenthesis A union B right parenthesis equals 5 over 6 comma P left parenthesis A intersection B right parenthesis equals 1 third and P left parenthesis A right parenthesis equals 2 over 3
    P(B)=P (AUB)- P(A) +P (A ∩ B) =5 over 6 minus 2 over 3 plus 1 third equals fraction numerator 5 minus 4 plus 2 over denominator 6 end fraction equals 3 over 6 equals 1 half
    As P(A). P(B)=2 over 3 cross times 1 half equals 1 third equals P left parenthesis A intersection B right parenthesis
    So, A and B are independent events.

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