Question

Problem 1) Let A, B and C be events from a common sample space such that: P(A) = 0.7, P(B) = 0.68, P(C) = 0.50, P(A∩B) = 0.42, P(A∩C) = 0.35, P(B∩C) = 0.34 (2 points) (i) Find P((A ∪ B) 0 ). (1 point) (ii) Find P(A|B). (2 points) (iii) Find P(A ∩ B ∩ C). (1 point) (iv) Are the events B and C independent? Justify your answer.

Answer 1

(i) Find P((A ∪ B) 0 ).

formula : P(A∪B)=P(A)+P(B)−P(A∩B)

P(A∪B)=0.7+0.68-0.42 = 0.96

(ii) Find P(A|B).

formula : P(A|B) = P(A∩B) /P(B)

= 0.42 / 0.68

= 0.6176

(iii) Find P(A ∩ B ∩ C).

formula : P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) − P(B ∩ C) + P(A ∩ B ∩ C)

and

P(A ∪ B ∪ C) = 1 {since A,B,C are all the vents in sample space}

1 = 0.7+0.68+0.5-0.42-0.35-0.34 + P(A ∩ B ∩ C)

P(A ∩ B ∩ C) = 1 - (0.7+0.68+0.5-0.42-0.35-0.34) = 0.23

P(A ∩ B ∩ C) = 0.23

(iv) Are the events B and C independent?

for them to be independent :

P(B)*P(C) = P(B∩C)

P(B∩C) = 0.34

and

P(B)*P(C) = 0.68*0.5 = 0.34

therefore,

P(B)*P(C) = P(B∩C) holds true

therefore,

events B and C are independent

(please UPVOTE)