Expected red balls in a scoop of five

Question

A box holds 20 balls, of which 12 are red and 8 are not. You scoop out 5 balls at random. On average, how many red balls are in your scoop?

Accepted Answer

Use linearity of expectation: find each red ball's chance of being scooped, then add. Take any one of the 12 red balls. A random scoop of 5 from 20 includes any given ball with probability \frac{5}{20}=\frac{1}{4}. Each red ball therefore contributes 	frac14 on average (1 if scooped, 0 if not). Add across all 12 reds: 12	imes\frac{1}{4}=3. Equivalently, sample size times the red fraction: 5	imes\frac{12}{20}=3. Takeaway. No matter how the draws depend on each other, the average count is just how many you draw times the share of the box that is red. Here five draws at a 60 percent red box g…

Expected red balls in a scoop of five

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