Variance of the number of red balls drawn without replacement

Question

A jar has 3 red and 3 blue balls. You draw 3 balls without replacement. Let X count how many of the drawn balls are red. What is the variance of X?

Accepted Answer

Think of a counter for each of the three draws: Ik = 1 if draw k is red. Because you do not put balls back, the draws are linked: pulling a red early leaves fewer reds for later. Each counter's own variance. By symmetry every draw is red with chance 	frac36 = 	frac12, so each counter has variance 	frac12\cdot	frac12 = 	frac14. Three of them give 3\cdot	frac14 = 	frac34. Covariance between two draws. Two specific draws are both red with chance 	frac36\cdot	frac25 = 	frac{6}{30} = 	frac15 (first red, then one fewer red out of one fewer ball). So \mathrm{Cov} = 	frac15 - 	frac12\cdo…

Variance of the number of red balls drawn without replacement

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