Ten balls are put in 6 slots at random. Then what is the expected total number of balls in the two extreme slots

Question

I am having the difficulty to predict the probability for this problem to find the total number of expectations.

Linearity of expectation! – Angina Seng May 02 '17 at 04:39 — Angina Seng, May 02 '17 at 04:39

score 3 · Answer 1 · answered May 02 '17 at 04:42

Let $X_i$ be the number of balls in bin $i$. Then $X_1 + X_2 + ... + X_{6} = 10$.

By linearity of expectation, we also have $E(X_1) + E(X_2) + ... + E(X_{6}) = 6E(X_1) = 10$, or $E(X_1) = \frac{10}{6} = 5/3$.

So in the two extreme bins, that's $E(X_1) + E(X_6) = 2E(X_1) = 10/3$

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