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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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P(B)=p1  P(B’)=1-p1

P(A∩B)=p2   P(A∩B’)=p3

now P(A)=P(A∩B)+P(A∩B’)=p2+p3 [ (A∩B) and (A∩B’) are mutually exclusive events]
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Given :

  • P(B)=p1
  • P(A $\cap$ B) =p2
  • P(A $\cap$ B’) =p3

 

We know that P(B/A)+P(B’/A)=1

$\Rightarrow$ P(B $\cap$ A)/P(A) + P(A $\cap$  B’)P(A) = 1
$\Rightarrow$ p2/P(A) + p3/P(A) =1  $\Rightarrow$ (p2+p3)/P(A)=1  $\Rightarrow$ P(A) = p2 + p3

 

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