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GATE 2021 CS (CS2) – Question 43

Engineering Mathematics · Probability and Statistics · 2 marks · Multiple choice

A bag initially has r red and b black balls. In each trial, draw a ball uniformly, note its colour, then return it with one additional ball of the same colour. Four trials are conducted. What is the probability that the fourth draw is red?

  1. r/(r+b)
  2. r/(r+b+3)
  3. (r+3)/(r+b+3)
  4. $\frac r{r+b}\frac{r+1}{r+b+1}\frac{r+2}{r+b+2}\frac{r+3}{r+b+3}$

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Correct answer: (A) r/(r+b)

Explanation

The fraction of red balls is a martingale: conditioning on the current counts, the expected red fraction after the replacement equals the current fraction. Consequently each next draw has the original marginal red probability r/(r+b). Thus A. D instead gives the probability that all four draws are red.

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