GATE CS 2026 Set 1 — Question 11
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Engineering Mathematics → Probability & Statistics → Conditional Probability
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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
Correct answer
(B) 1/3
Solution
Let denote the number of red and blue balls in the urn. Initially, the state is . Total balls = 2.Step 1:
We want the final state to have an equal number of red and blue balls. Since we start with 2 balls and add 1 ball in each of the 2 steps, the final total is 4 balls. Thus, we need the final state to be .
Probability of this path = .
Probability of this path = .Total probability = .
- Probability of picking Red () = . New state: . Total balls = 3.
- Probability of picking Blue () = . New state: . Total balls = 3.
We want the final state to have an equal number of red and blue balls. Since we start with 2 balls and add 1 ball in each of the 2 steps, the final total is 4 balls. Thus, we need the final state to be .
- Case 1: From state (after picking Red in Step 1):
Probability of this path = .
- Case 2: From state (after picking Blue in Step 1):
Probability of this path = .Total probability = .
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