Question

Two boxes are labelled I and II. Box I contains 4 black balls and 6 white balls. Box II contains 2 black balls and 8 white balls. A player randomly selects one box and then randomly draws 2 balls without replacement from the selected box.

1) What is the probability of obtaining 2 black balls?

2) Given that 2 black balls are obtained, what is the probability that Box I is selected?

3) Suppose that 20 players conduct the game independently. What is the probability that exactly 4 players obtain 2 black balls?

Answer 1

B1 : Event of selecting Box 1

B2 : Event of selecting Box 2

P(B1)=P(B2)=1/2

X: Event of obtaining 2 black balls

P(X|B1) = probability of obtaining 2 black balls given that Box 1 was selected

Number of black balls in Box 1 = 4

Number white balls in Box 1 = 6

Total number of balls in Box 1 = 10

probability of obtaining 2 black balls given that Box 1 was selected

= Number of ways of selecting 2 black balls from 4 black balls / Number of ways of selecting 2 balls from 10 balls

P(X|B1) = 6/45

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X: Event of obtaining 2 black balls

P(X|B2) = probability of obtaining 2 black balls given that Box 2was selected

Number of black balls in Box 2 = 2

Number white balls in Box 2 = 8

Total number of balls in Box 2 = 10

probability of obtaining 2 black balls given that Box 2 was selected

= Number of ways of selecting 2 black balls from 2 black balls / Number of ways of selecting 2 balls from 10 balls

P(X|B2) = 1/45

1)

Probability of obtaining 2 black balls = P(X)

P(X)=P(B1)P(X|B1)+P(B2P(X|B2) = (1/2) x (6/45) + (1/2) x (1/45) = 6/90 + 1/90 = 7/90 = 0.077777778

Probability of obtaining 2 black balls = 7/90 = 0.077777778

2) Given that 2 black balls are obtained, probability that Box I is selected = P(B1|X)

By Bayes theorem,

Given that 2 black balls are obtained, probability that Box I is selected = 6/7 = 0.857142857

3)

Suppose that 20 players conduct the game independently. probability that exactly 4 players obtain 2 black balls

Y : Number of players obtain 2 black balls

Number of players : n=20

p: Probability that a randomly selected player obtain 2 black balls = P(X)= (7/90) (From 1)

Y follows binomial distribution with n=20 and p=(1/45) ; q =1-p=1-(7/90) = (83/90)

Probability mass function of Y :

Probability that 'r' players out of 20 players obtain 2 black balls = P(Y=r)

Probability that exactly 4 players obtain 2 black balls = P(Y=4)

Probability that exactly 4 players obtain 2 black balls = 0.0485384