1 In an experiment C D are independent events with probabili

1) In an experiment, C, D are independent events with probabilities P[ intersection D] = 1/3, P[C] = 1/2

a) Find P[D], P[C intersection D^(c)], P[C^(c) intersection D^(c)]

b) Find P[C union D] and P[C union D^(c)]

c) Are C and D^(c) independent?

2) For the independent events A and B, prve that

a) A and B^(c) are independent

b) A^(c) and B are independent

c) A^(c) and B^(c) are independent

Solution

1.)

a.) C and D are independent means P(C intersection D) = P(C)*P(D)

So 1/3 = (1/2)*P(D)

So P(D) = 2/3

P(C intersection D^c) = P(C)*(1-P(D)) = (1/2)*(1/3) = 1/6

P(C^c intersection D^c) = (1-P(C))*(1-P(D)) = (1/2)*(1/3) = 1/6

b.) P(C union D) = P(C) + P(D) - P(C intersection D) = 1/2 + 2/3 - (1/2)*(2/3) = 5/6

P(C union D^c) = P(C) + 1 - P(D) - P(C intersection D^c) = 1/2 + 1/3 - 1/6 = 2/3

c.) P(C intersection D^c) = P(C)*P(D^c) = 1/6

So C and D^c are independent

2.) P(A intersection B) = P(A)*P(B)

a.) P(A)P(B^c) = P(A)(1-P(B)) = P(A) - P(A intersection B)

= P(A intersection B^c)

So A and B^c are independent.

b.) P(A^c)P(B) = (1-P(A))P(B) = P(B) - P(A intersection B)

= P(A^c intersection B)

So A^c and B are independent.

c.) P(A^c)P(B^c) = (1-P(A))(1-P(B)) = 1 - P(A) - P(B) + P(A intersection B) = 1 - P(A union B) = P((A union B)^c)

= P(A^c intersection B^c)

So A^c and B^c are independent.