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GATE CSE 2026 Session 1 Q11: Pólya urn: equal red and blue after two steps

Question

An urn contains one red ball and one blue ball. At each step, a ball is picked uniformly at random from the urn, and this ball together with another ball of the same color is put back in the urn. The probability that there are equal number of red and blue balls after two steps is

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Answer: B (official GATE 2026 final answer key)

Concept used: Sequential conditional probability (Pólya's urn).

First step: After step 1 the urn is 2–1 either way; equality needs the minority colour on step 2, with probability 1/31/3.

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Source: official GATE 2026 Session 1 CS question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key

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