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NDA 2020 Maths · Mathematics · Probability · hard

Urn U1 contains 4 red and 2 black balls, and urn U2 contains 1 red and 5 black balls. An urn is chosen at random and a ball drawn from it is red. The…

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Urn U1U_1 contains 44 red and 22 black balls, and urn U2U_2 contains 11 red and 55 black balls. An urn is chosen at random and a ball drawn from it is red. The probability that it came from U1U_1 is
  1. A. 23\dfrac23
  2. B. 12\dfrac12
  3. C. 45\dfrac45
  4. D. 56\dfrac56
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Answer: (C) 45\dfrac45

By Bayes' theorem, P(U1∣R)=12⋅2312⋅23+12⋅16=1313+112=4/125/12=45P(U_1\mid R)=\dfrac{\frac12\cdot\frac23}{\frac12\cdot\frac23+\frac12\cdot\frac16}=\dfrac{\frac13}{\frac13+\frac1{12}}=\dfrac{4/12}{5/12}=\dfrac45. The value 23\dfrac23 is only P(R∣U1)P(R\mid U_1), the likelihood, which is not the posterior probability asked for. The value 12\dfrac12 is the prior before observing the colour, and 56\dfrac56 is P(black∣U2)P(\text{black}\mid U_2).
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Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake