JEE Advanced 2018 Paper 1 · Mathematics · Probability · medium

Urn UA contains 3 red and 2 blue balls and urn UB contains 2 red and 3 blue balls. A fair die is rolled; if the number is 1 or 2 the urn UA is…

One or more options correct

Urn

U_A

contains

3

red and

2

blue balls and urn

U_B

contains

2

red and

3

blue balls. A fair die is rolled; if the number is

1

2

the urn

U_A

is chosen, otherwise

U_B

is chosen. Two balls are then drawn from the chosen urn one after the other without replacement. Which of the following is (are) correct?

A. The probability that both balls drawn are red is $\dfrac16$
B. Given that both balls drawn are red, the probability that the urn was $U_A$ is $\dfrac35$
C. Given that the first ball drawn is red, the probability that the second ball is blue is $\dfrac23$
D. The probability that both balls drawn have the same colour is $\dfrac3{10}$

Show answer and solution

Answer: A, B

Here

P(U_A)=\frac13

P(U_B)=\frac23

. For both red:

U_A

gives

\frac35\cdot\frac24=\frac3{10}

and

U_B

gives

\frac25\cdot\frac14=\frac1{10}

, so

P(RR)=\frac13\cdot\frac3{10}+\frac23\cdot\frac1{10}=\frac16

(A true). By Bayes,

P(U_A\mid RR)=\frac{1/10}{1/6}=\frac35

(B true).

P(\text{first red})=\frac13\cdot\frac35+\frac23\cdot\frac25=\frac7{15}

and

P(R\text{ then }B)=\frac13\cdot\frac3{10}+\frac23\cdot\frac3{10}=\frac3{10}

, so the conditional probability of a blue second ball is

\frac{9}{14}\neq\frac23

(C false).

P(BB)=\frac13\cdot\frac1{10}+\frac23\cdot\frac3{10}=\frac7{30}

, so

P(\text{same colour})=\frac16+\frac7{30}=\frac25\neq\frac3{10}

(D false).

Original question written in the JEE Advanced 2018 Paper 1 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake