In a bag of 50 balls, 15 are red, 20 are blue, and 15 are green. If 5 balls are drawn at random without replacement, what is the probability that exactly 3 of them are red?

Instruction: Use the hypergeometric distribution formula to calculate this probability.

Context: This question evaluates the candidate's understanding of the hypergeometric distribution and its application in a drawing without replacement scenario.

Updated

Official answer

Choose three of 15 red balls and two of 35 non-red balls. For a uniform sample of five distinct balls,

P(exactly three red) = C(15,3) C(35,2) / C(50,5)

= 455 × 595 / 2118760 = 270725/2118760 ≈ 0.127775, or 12.78%.

The other two balls must be non-red to enforce “exactly.” The denominator counts unordered five-ball samples, so the numerator uses the same convention. A binomial model with p=15/50 would describe independent draws with replacement and is not the exact distribution here.

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