Expected cracked eggs grabbed from a crate

Question

A crate holds 24 eggs, of which 6 are cracked and 18 are intact. You grab 8 eggs at random. On average, how many cracked eggs do you grab?

Accepted Answer

Use linearity of expectation: find each cracked egg's chance of being grabbed, then add. Take any one of the 6 cracked eggs. A random grab of 8 from 24 includes any given egg with probability \frac{8}{24}=\frac{1}{3}. Each cracked egg contributes 	frac13 on average. Add over all 6: 6	imes\frac{1}{3}=2. Or directly, draws times target fraction: 8	imes\frac{6}{24}=2. Takeaway. The average number of a type you pull equals the number of draws times that type's share of the pile. A quarter of the crate is cracked and you grabbed eight, so two cracked on average.

Expected cracked eggs grabbed from a crate

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