Question

3. Box A contains 6 red balls and 3 green balls, whereas box B contains 3 red ball and 15 green balls.

Stage one:One box is selected at random in such a way that box A is selected with probability 1/5 and box B is selected with probability 4/5.

Stage two: First, suppose that 1 ball is selected at random from the box selected at stage one.

a) What is the probability that the ball is red?

b) Given that the ball is red, what is the probability it came from box A? Next, suppose that two balls are selected at random without replacement from the box selected at stage one.

c) What is the probability that both balls are red?

d) Given that both balls are red, what is the probability they came from box A?

e) What is the probability that one ball is red and the other is green?

f) Given that one ball is red and the other is green, what is the probability they came from box A?

Answer 1

3.

Box A :

Number of red balls = 6

Number of green balls = 3

Box B :

Number of red balls = 3

Number of green balls = 15

A : Event of selecting Box A

B : Event of selecting Box B

P(A) = 1/5 ; P(B) =4/5

a)

R : Event of selecting red Ball

G : Event of selecting Green ball

P(R|A) = Probability of selecting Red Ball given that Box A is selected = Number of red balls in Box A / Total number of balls in Box A= 6/9

P(R|B) = Probability of selecting Red Ball given that Box B is selected = Number of red balls in Box B / Total number of balls in Box B= 3/18

suppose that 1 ball is selected at random from the box selected at stage one, probability that the ball is red

P(R) = P(A)P(R|A)+P(B)P(R|B) = (1/5)x(6/9) + (4/5)x(3/18) = 6/45 + 6/45 = 12/45

suppose that 1 ball is selected at random from the box selected at stage one, probability that the ball is red = 12/45

b)

Given that the ball is red, probability it came from box A = P(A|R)

Given that the ball is red, probability it came from box A = 0.5

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c)

Next, suppose that two balls are selected at random without replacement from the box selected at stage one

R2 : Event of selecting both red Ball

P(R2|A)

= Probability of selecting both red Balls given that Box A is selected

= Number of ways 2 red balls from 6 red balls in Box A / Number of ways of selecting 2 balls from 9 balls

P(R2|B)

= Probability of selecting both red Balls given that Box B is selected

= Number of ways 2 red balls from 3 red balls in Box B/ Number of ways of selecting 2 balls from 18 balls

P(R|B) = Probability of selecting Red Ball given that Box B is selected = Number of red balls in Box B / Total number of balls in Box B= 3/18

Probability that both balls are red = P(R2)

P(R2) = P(A)P(R2|A)+P(B)P(R2|B) = (1/5)x(5/12) + (4/5)x(3/153) = 1/12+12/765 = (765+144)/9180=909/9180

Probability that both balls are red = P(R2) = 909/9180

d) Given that both balls are red, probability they came from box A = P(A|R2)

Given that both balls are red, probability they came from box A = 765/909

e)

Next, suppose that two balls are selected at random without replacement from the box selected at stage one

RG : Event of selecting one Ball is red and other is green

P(RG|A)

= Probability of selecting one Ball is red and other is greengiven that Box A is selected

= Number of ways 1 red balls from 6 red balls and 1 green ball from 3 green balls in Box A / Number of ways of selecting 2 balls from 9 balls

P(RG|B)

= Probability of selecting one Ball is red and other is greengiven that Box B is selected

= Number of ways 1 red balls from 3 red balls and 1 green ball from 15 green balls in Box A / Number of ways of selecting 2 balls from 18 balls

probability that one ball is red and the other is green = P(RG)

P(RG) = P(A)P(RG|A)+P(B)P(RG|B) = (1/5)x(1/2) + (4/5)x(5/17) = 1/10+4/17 = (17+40)/170=57/170

probability that one ball is red and the other is green = 57/170

f) Given that one ball is red and the other is green, the probability they came from box A = P(A|RG)

Given that one ball is red and the other is green, the probability they came from box A = 17/57