GATE CS 2021 Set 2 Q.43 (CS Q.33): Probability and Statistics

A bag has $r$ red balls and $b$ black balls. All balls are identical except for their colours. In a trial, a ball is randomly drawn from the bag, its colour is noted and the ball is placed back into the bag along with another ball of the same colour. Note that the number of balls in the bag will increase by one, after the trial. A sequence of four such trials is conducted. Which one of the following choices gives the probability of drawing a red ball in the fourth trial?

$\dfrac{r}{r+b}$

$\dfrac{r}{r+b+3}$

$\dfrac{r+3}{r+b+3}$

$\left(\dfrac{r}{r+b}\right) \left(\dfrac{r+1}{r+b+1}\right) \left(\dfrac{r+2}{r+b+2}\right) \left(\dfrac{r+3}{r+b+3}\right)$

Official answer and solution

Official answer: A
From: Official GATE 2021 answer key

Solution

Apply the invariance property of Polya's Urn model under reinforcement sampling.

The bag initially contains $r$ red balls and $b$ black balls, for a total of $r + b$ balls. In each trial, a ball is drawn, observed, and returned with one additional ball of the same colour. This is the classical Polya Urn model with parameter $c = 1$ .

In Polya's urn scheme, the marginal probability of drawing a red ball at any given trial $k$ (where $k = 1, 2, 3, 4, \dots$ ) is strictly invariant and equals the initial proportion of red balls in the urn:

P(R_k) = \frac{r}{r+b}

To see this explicitly for trial 2:

P(R_2) = P(R_1) P(R_2 \mid R_1) + P(B_1) P(R_2 \mid B_1)

P(R_2) = \frac{r}{r+b} \times \frac{r+1}{r+b+1} + \frac{b}{r+b} \times \frac{r}{r+b+1} = \frac{r(r+1) + br}{(r+b)(r+b+1)} = \frac{r(r+b+1)}{(r+b)(r+b+1)} = \frac{r}{r+b}

By induction, this invariance holds for all trials, including the fourth trial.

Thus, the probability of drawing a red ball in the fourth trial is $\dfrac{r}{r+b}$ , which corresponds to option A.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2021 Computer Science and Information Technology - Session 2 (CS2)
Question: 43 of 65, printed as CS Q.33
Section: Computer Science and Information Technology

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