JEE Advanced 2021 Paper 1 · Mathematics · Probability · medium

A fair die is rolled once. If the number shown is 1 or 2, two balls are drawn together (without replacement) from bag A containing 3 red and 2 blue…

Single correct option

A fair die is rolled once. If the number shown is

1

2

, two balls are drawn together (without replacement) from bag

A

containing

3

red and

2

blue balls; otherwise two balls are drawn together from bag

B

containing

2

red and

4

blue balls. Given that the two balls drawn are of the same colour, the probability that they came from bag

A

Show answer and solution

Answer: (B) $\dfrac{3}{10}$

Use Bayes' theorem. Bag

A

is chosen with probability

\tfrac13

(die shows 1 or 2) and bag

B

with probability

\tfrac23

. From

A

, P(both red)

=\tfrac{3\cdot2}{5\cdot4}=\tfrac{3}{10}

and P(both blue)

=\tfrac{1}{10}

, so P(same colour

\mid A

)

=\tfrac25

. From

B

, P(both red)

=\tfrac{2\cdot1}{6\cdot5}=\tfrac1{15}

and P(both blue)

=\tfrac{4\cdot3}{6\cdot5}=\tfrac25

, so P(same colour

\mid B

)

=\tfrac7{15}

. Then

P(A\mid \text{same})=\dfrac{\frac13\cdot\frac25}{\frac13\cdot\frac25+\frac23\cdot\frac7{15}}=\dfrac{6/45}{6/45+14/45}=\dfrac{3}{10}

. The options

2/5

and

7/15

are the conditional probabilities within a single bag, forgetting the prior weights.

Original question written in the JEE Advanced 2021 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