Question

Problem 3. A ball is taken at random from an urn containing 2 red balls and 3 blue balls. If the ball is red a fair coin is tossed three times. If the ball is blue non-fair coin is tossed three times; for this second coin the probability of heads is .6. In either case we count the number of heads in the three tosses and call that number X (a) Compute the conditional probability that X-2 if it is knoun that the selected ball is red b) Compute the conditional probability that X-2 if it is knoun that the selected ball is blue (c) Compute the probability that X-2. (d) Compute the conditional probability that the selected ball is blue given that X2

Answer 1

a)P(X=2|red)=P(2 heads out of 3 toss of fair coin)=³C₂(0.5)²(0.5)¹ =0.375

b)P(X=2|blue)=P(2 heads out of 3 toss of non-fair coin)=³C₂(0.6)²(0.4)¹ =0.432

c)

P(X=2)=P(red)*P(X=2|red)+P(blue)*P(X=2|blue)

=(2/5)*0.375+(3/5)*0.432=0.4092

d)

P(blue|X=2)=P(blue)*P(X=2|blue)/P(X=2)=(3/5)*0.432/0.4092=0.6334