GATE CS 2021 Set 1 Q.8 (GA Q.8): Quantitative Aptitude

There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag.

The probability that at least two chocolates are identical is

0.3024

0.4235

0.6976

0.8125

Official answer and solution

Official answer: C
From: Official GATE 2021 answer key

Solution

Use complementary probability: subtract the probability that all 5 chosen chocolates are distinct from 1.

Each of the 5 bags contains 10 distinct chocolates. One chocolate is drawn independently from each bag, giving a total of $10^5 = 100000$ possible outcomes.

The event that all 5 selected chocolates are mutually distinct means: the first bag chocolate can be any of 10, the second bag choice must be different (9 choices), the third must differ from the first two (8 choices), the fourth has 7 choices, and the fifth has 6 choices.

The number of distinct selections is:

10 \times 9 \times 8 \times 7 \times 6 = 30240

P(\text{all distinct}) = \frac{30240}{100000} = 0.3024

The probability that at least two chocolates are identical is the complement:

P(\ge 2\text{ identical}) = 1 - P(\text{all distinct}) = 1 - 0.3024 = 0.6976

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 (CS, Set-1)
Question: 8 of 65, printed as GA Q.8
Section: General Aptitude

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