JEE Advanced 2024 Paper 2 · Mathematics · Probability · medium

A fair die is thrown once. If the result is 1 or 2, two balls are drawn without replacement from Urn I containing 3 red and 2 blue balls; otherwise…

Single correct option

A fair die is thrown once. If the result is

1

2

, two balls are drawn without replacement from Urn I containing

3

red and

2

blue balls; otherwise two balls are drawn without replacement from Urn II containing

2

red and

4

blue balls. Given that the two balls drawn are of the same colour, the probability that Urn I was used is

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

P(\text{same colour}\mid I)=\dfrac{\binom32+\binom22}{\binom52}=\dfrac{4}{10}=\dfrac25

and

P(\text{same}\mid II)=\dfrac{\binom22+\binom42}{\binom62}=\dfrac{7}{15}

. With

P(I)=\tfrac13

P(II)=\tfrac23

P(I\cap\text{same})=\tfrac{2}{15}

and

P(II\cap\text{same})=\tfrac{14}{45}

. By Bayes' theorem

P(I\mid\text{same})=\dfrac{6/45}{6/45+14/45}=\dfrac{3}{10}

. The value

\frac{2}{15}

is the joint probability (no normalisation) and

\frac13

is the prior.

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