Question

Q. Let A, B independent events, with P(A) = 1/2 and P(B) = 2/3. Now C be an event with P(C) = 1/4, and suppose that P(A|C) = 1/3, P(B|?̅) =7/9, P(A∩B|?̅) = 7/18.
(a) Calculate the P(A∩B)
(b) Calculate the P(A|?̅) and P(B|C)
(c) Calculate the P(A∩B|C)
(d) Show if P(A∩B|C) equals P(A|C)P(B|C) or not.

Answer 1

Answer:

Given,

P(A) = 1/2

P(B) = 2/3

P(C) = 1/4

P(A|C) = 1/3

P(B|C) = 7/9

P(A ∩ B|C) = 7/18

a)

P(A ∩ B) = P(A)*P(B) [since A & B are independent]

= 1/2 * 2/3

= 1/3

b)

P(A|C') & P(B|C)

consider,

P(A|C') = P(A ∩ C') / P(C')

= [P(A) - P(A ∩ C)] / [1-P(C)]

= [1/2 - 1/12] / [1 - 1/4] [since P(A ∩ C = P(C)*P(A|C) = 1/3*1/4 = 1/12]

P(A|C') = 5/9

Now [since P(B|C') = 7/9 ; i.e.,

P(B ∩ C')/P(C') = 7/9 ;

P(B) - P(B ∩ C) = 7/9*P(C') ;

P(B ∩ C)= - [ 7/9(1-1/4) - 2/3] = 1/12]

P(B|C) = P(B ∩ C)/P(C)

= (1/12) / (1/4)

P(B|C) = 1/3

C)

[consider,

P(A ∩ B|C') = 7/18

P(A ∩ B ∩ C')/P(C') = 7/18

P(A ∩ B) - P(A ∩ B ∩ C) = 7/18(1-1/4)

-P(A ∩ B ∩ C) = 7/24 - P(A ∩ B)

P(A ∩ B ∩ C) = P(A ∩ B) - 7/24] ---------> (1)

P(A ∩ B|C) = P(A ∩ B ∩ C)/P(C)

substitute (1) in above formula

= [P(A ∩ B) - 7/24] / (1/4)

= [1/3 - 7/24]/(1/4)

= 1/24 / 1/4

P(A ∩ B|C) = 1/6

d)

P(A|C)*P(B|C) = 1/3 * 1/3

= 1/9

So P(A ∩ B|C) != P(A|C)*P(B|C)