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Let $P(E)$ denote the probability of the occurrence of event $E$. If $P(A)= 0.5$ and $P(B)=1$ then the values of $P(A|B)$ and $P(B|A)$ respectively are

1. $0.5, 0.25$
2. $0.25, 0.5$
3. $0.5, 1$
4. $1, 0.5$
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+1
Since P(B)=1, event B denotes the sample space and event A where P(A)<1, denotes an event...

Hence A intersection B = A

P(A)= 0.5

P(B)=1

Here there is no dependency in event A and B.
So  P(A $\cap$ B) = P(A) * P(B)

P(A/B)= probability of occurrence  of event A when B has already occurred

= P(A $\cap$ B) / P(B)

= (0.5 * 1) /1 = 0.5

P(B/A)= probability of occurrence  of event B when A  has already occurred

= P(B $\cap$ A) / P(A)

= (1 * 0.5) / 0.5 =  1

Ans- C

by Boss (26.6k points)
edited by
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HOW U KNOW THERE IS NO DEPENDENCY
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@ashutosh , Since there is nothing said about dependency so I assumed both A and B are  independent event.
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P(A/B)=$\frac{P(A)*P(B/A)}{P(B)}$

P(A/B)=0.5 * P(B/A)

0.25=0.5 * 0.5

If they are dependent option B will be the answer?
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C also satisfiy your equation then how could you say that only B is the option...if they are dependent then ...

P(B) would be less that 1...

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Thank you.Can you please clear it again?

I am not saying option B is answer. I am asking if A and B are dependent (say given in question) then can option B be the answer(and not C)? This is my doubt.
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NOTE. IF SOMEONE THINKS LIKE FOLLOWING WAYS, THEN HE/SHE IS LACKING CONCEPT OF PROBABILITY.

BY FOLLOWING WAYS, ANSWER IS CORRECT, BUT METHOD IS ABSOLUTELY WRONG.

We know from set theory A INTERSECTION B =  A-(A-B)

Here, SET B CONTAINS MORE ELEMENT THAN SET A as P(B)=1 and P(A)=0.5

Therefore, A INTERSECTION B = B-(B-A) = 1-(1-0.5)= 1-0.5= 0.5

now, substituting value of A INTERSECTION B in respective equations we can get the answers. using

P(B/A)= P(A INTERSECTION B)/P(A) as A INTERSECTION B = B INTERSECTION A ( from set theory property)

Hence, option C is the correct option.