GATE CS 2021 Set 2 — Question 43
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Engineering Mathematics → Probability & Statistics → Conditional Probability
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Question
A bag has red balls and 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?
Correct answer
(A) (r)/(r+b)
Solution
This problem describes Pólya's Urn model. In this model, the probability of drawing a ball of a specific color in any trial is equal to the initial probability of drawing that color in the first trial.
Initially, .
For the second trial:
.
By induction, the probability of drawing a red ball in any trial remains . Thus, for the fourth trial, the probability is .
Initially, .
For the second trial:
.
By induction, the probability of drawing a red ball in any trial remains . Thus, for the fourth trial, the probability is .
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