Counting, total probability, and independence

Problem

A class has 66 students, 44 of whom use Python. Two students are selected uniformly without replacement. Find the probability that both use Python. Then state whether the two selection indicators are independent.

Reveal answer or reference solution

The probability is

(42)(62)=615=25.\frac{\binom{4}{2}}{\binom{6}{2}} = \frac{6}{15} = \frac{2}{5}.

The indicators are not independent because sampling without replacement changes the second probability after the first outcome.

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