JEE Advanced 2024 Paper 1 · Mathematics · Probability · medium

An urn UA has 3 red and 2 blue balls, and an urn UB has 2 red and 4 blue balls. A fair die is rolled; if the number is 1 or 2, urn UA is chosen…

Single correct option

An urn

U_A

has 3 red and 2 blue balls, and an urn

U_B

has 2 red and 4 blue balls. A fair die is rolled; if the number is 1 or 2, urn

U_A

is chosen, otherwise

U_B

is chosen. Two balls are then drawn without replacement from the chosen urn, and both are found to be of the same colour. The probability that urn

U_A

was chosen is

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

Let

E

be the event that the two balls have the same colour. For

U_A

P(RR)=\dfrac35\cdot\dfrac24=\dfrac{3}{10}

and

P(BB)=\dfrac25\cdot\dfrac14=\dfrac1{10}

, so

P(E\mid A)=\dfrac25

. For

U_B

P(RR)=\dfrac26\cdot\dfrac15=\dfrac1{15}

and

P(BB)=\dfrac46\cdot\dfrac35=\dfrac25

, so

P(E\mid B)=\dfrac{7}{15}

. The prior probabilities are

P(A)=\dfrac13

and

P(B)=\dfrac23

. By Bayes' theorem

P(A\mid E)=\dfrac{\frac13\cdot\frac25}{\frac13\cdot\frac25+\frac23\cdot\frac7{15}}=\dfrac{2/15}{2/15+14/45}=\dfrac{6/45}{20/45}=\dfrac{3}{10}

. The value

\frac13

is only the prior,

\frac25

is the likelihood

P(E\mid A)

alone, and ignoring the unequal prior of the two urns would give a different value.

Original question written in the JEE Advanced 2024 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