MAD1100 Probabilities Sets A nickel a dime and a quarter are

A nickel, a dime, and a quarter are each tossed once. Assume each coin is balanced. Let C be the event \"Exactly two heads come up.\" Let D be the event \"At. least two heads come up.\" Find P(C|D) and P(D|C). Are the events C and D independent?

Solution

Making cases of Head and Tails on each coin

H H H

H H T

H T H

H T T

T T T

T H T

T T H

T H H

P(C) = (No. of Cases with Exactly Two Heads)/(Total No. of Cases) = 3/8

P(D) = (No. of Cases with Atleast Two Heads)/(Total No. of Cases) = 4/8

P(CD) = (No. of Cases with Exactly Two Heads)/(Total No. of Cases) = 3/8

P(C|D) = P(CD)/P(D) = (3/8) / (4/8) = 3/4

P(D|C) = P(CD)/P(C) = (3/8) / (3/8) = 1

P(C) * P(D) = (3/8) * (4/8) = 3/16 P(CD)

Therefore , C and D are not independent