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25 votes
25 votes
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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Best answer
33 votes
33 votes
$P(A ∩ B') = P(A) - P(A ∩ B)$

$\implies P(A)$ $=$  $P_{2}+P_{3}$
edited by
7 votes
7 votes
Use Law of Total Probability:

$P(\text{A}) = P(\text{A and B}) +P(\text{P and Not B})$

$\Rightarrow P(\text{A}) = P(A \cap B) +P(P \cap \overline{B})$

$\Rightarrow P(\text{A}) = P_2 + P_3$
0 votes
0 votes

Well, from venn diagram it looks pretty obvious that the 

P(A) = P1 + P2

But, I tried another approach...

@Arjun sir, can you please check this out..

P(A) = P(A/B) + P(A/B')

That is, the probability of occurrence of A is the sum of:

i) Probability that A occurs given that B has occurred: P(A/B)
ii) Probability that A occurs given that B' has occurred: P(A/B')

Since, B and B' totally make up sample space, the occurrence of A is also covered.

P(A/B) = P(A∩B)  =  P2
                 P(B)            P1

P(A/B') = P(A∩B')  =  P3   
                  P(B')          1-P1

P(A) = P2   +   P3   
             P1       1-P1

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