GATE CS 2024 Set 1 Q.63: Probability and Statistics

A bag contains 10 red balls and 15 blue balls. Two balls are drawn randomly without replacement. Given that the first ball drawn is red, the probability (rounded off to 3 decimal places) that both balls drawn are red is _________

Your answer

Official answer and solution

Accepted answers: 0.37 to 0.38
From: Official GATE 2024 answer key

Solution

Condition on the first ball being red: compute the fraction of red balls remaining among the remaining 24 balls.

Initially, the bag has $10$ red balls and $15$ blue balls ( $25$ balls in total). Given that the first ball drawn is red (and drawn without replacement), the bag has $10 - 1 = 9$ red balls and $15$ blue balls remaining ( $24$ balls in total). Therefore, the conditional probability that the second ball is also red (so that both balls drawn are red) is:

P(R_1 \cap R_2 \mid R_1) = P(R_2 \mid R_1) = \frac{9}{24} = \frac{3}{8} = 0.375

Final answer

0.375

Common trap

Computing the unconditional probability $\frac{10}{25} \times \frac{9}{24} = 0.15$ instead of the conditional probability given the first ball is red.

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

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 1 (CS1)
Question: 63 of 65
Section: Computer Science and Information Technology

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