Question

If P(AUB) = P(ANB) for any two events A and B,
then
P(A)=P(B)
P(A) <PB)
P(A)> PB)
None of these​

Answer 1

Given  :  P(AUB) = P(A∩B) for any two events A and B

To Find : Correct option

P(A)=P(B)

P(A) <PB)

P(A)> PB)

None of these​

Solution:

P(AUB)  = P(A) + P(B)  - P(A∩B)

=>  P(A) + P(B)  =  P(AUB) +  P(A∩B)

P(AUB) = P(A∩B)

=>  P(A) + P(B)  =  P(AUB) +  P(AUB)  

as P(AUB)   ≥  P(A)  , P(B)

This  is only possible when

P(AUB)  = P(A)   and P(AUB)  = P(B)

Hence  

P(A)  = P(B)

Correct  option is  P(A)  = P(B)

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Answer 2

SOLUTION

TO CHOOSE THE CORRECT OPTION

If P(AUB) = P(A∩B) for any two events A and B,

then

P(A) = P(B)

P(A) < P(B)

P(A) > P(B)

None of these

EVALUATION

Here it is given that

P(AUB) = P(A∩B)

We are aware of the formula on probability that

P(AUB) = P(A) + P(B) - P(A∩B)

⟹ P(A∩B) = P(A) + P(B) - P(A∩B)

⟹ P(A∩B) - P(A) = P(B) - P(A∩B) - - - - - (1)

Again we know that

P(A∩B) ≤ P(A) and P(A∩B) ≤ P(B)

So the Expression 1 is possible only if

P(A∩B) = P(A) = P(B)

FINAL ANSWER

Hence the correct option is

P(A) = P(B)

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