A For three events A B and C we know that i A and C are inde

A. For three events A, B, and C, we know that i. A and C are independent, ii. B and C are independent, iii. A and B are disjoint, iv. P(AC)=2/3,P(BC)=3/4,P(ABC)=11/12 find P(A),P(B),P(C)

Solution

A and C are independent ->   P(AUC) =P(A) + P(C) -P(A and C) = P(A) +P(C) - P(A)*P(C)= 2/3

B and C are independent ->   P(BUC) =P(B) + P(C) -P(B and C) = P(B) +P(C) -P(B)*P(C)= 3/4

A and B are disjoint ->   P(AUB) =P(A) + P(B) -P(A and B) = P(A) +P(B)

P(AUBUB) = P(A) + P(B) + P(C) - P(A and B) - P(B and C) -P(A and C) + 2P(A and B and C)

               =P(A) + P(B) + P(C) - P(B)* P(C) -P(A) * P(C) =11/12

      -> 2/3 +3/4 -P(C) =11/12

    -> P(C) = 2/3 +3/4 -11/12= 6/12= 1/2

P(A) +P(C) - P(A)*P(C)= 2/3 -> P(A) = [2/3-1/2]*2 = 1/3

P(B) +P(C) -P(B)*P(C)= 3/4 -> P(B) = [3/4-1/2]*2 = 1/2