A random number of apples, each of random weight

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A basket is filled with N apples, where N is equally likely to be 1, 2, or 3. Each apple, independently, weighs a random amount that averages 150 grams. What total weight should you expect for the basket?

Accepted Answer

The number of apples is random, and each apple's weight is random. The tower rule keeps the two layers apart: E[	ext{weight}] = E\big[\,E[	ext{weight} \mid N]\,\big]. Step 1, freeze the count. If you know N = n, the basket holds n apples, each averaging 150 grams, so E[	ext{weight} \mid N = n] = 150\,n. Step 2, average over the count. The count is uniform on \{1,2,3\}, so E[N] = \frac{1+2+3}{3} = 2. Then E[	ext{weight}] = E[150\,N] = 150\,E[N] = 150 	imes 2 = 300 	ext{ grams}. So you expect 300 grams: expected number of apples (2) times average weight per apple (150). Takeaway. Even whe…

A random number of apples, each of random weight

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