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The probability of an event $B$ is $P_1$. The probability that events $A$ and $B$ occur together is $P_2$ while the probability that $A$ and $\bar{B}$ occur together is $P_3$. The probability of the event $A$ in terms of $P_1, P_2$ and $P_3$ is _____________

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we know ,P(B | A)+ P(B’|A) = 1

given,

P(B’|A) P(A) = p3

P(B|A) P(A) = p2...(1)

P(B|A) = 1-(p3/P(A))

So, P(B|A) P(A) = P(A) -p3……...(2)

and from 1 and 2

P(A) = p2+p3.

We can use set theoretic approach to solve it:

P(A  intersection  Bcomplement ) = P(A  U  B) +  P(B) {you can check through Venn representation for more clarity}

and now P(A  U  B) = P(A) + P(B) – P(A intersection B)

So,

P(A  intersection  Bcomplement ) = P(A) + P(B) – P(A intersection B) +  P(B)

Putting values given according to question will yield equation as

P3 = P(A) + P1 – P2 – P1

Therefore, P(A) = P3 + P2

3p2+p1-p2

Wrong

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