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Suppose you roll three fair 100-sided die. What is the expected value of the lowest roll?

Someone explained their solution to me but I am having trouble with wrapping my head around the intuition in particular, why can we rewrite the min to $(100^3 + 99^3 + ... + 1^3)$ and how do we go from $(100^3 + 99^3 + ... + 1^3)/(100^3) \rightarrow (1 + 2 + ... + 100)^2$. What is expectation by survival? It's starting to make sense somewhat when I look at the situation for 1 rolling of 6-sided die but I want to hear if anyone has an intuitive way of thinking about this solution. Let me know if anything is unclear or you have questions. Would also like to hear any additional solutions if they are efficient.

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I believe what your friend means by "survival" is, for a non-negative discrete random variable $X$, it holds $$E[X] = \sum_{k\geq 0} P(X \geq k).$$ (See this post). In this case, let $X_1,X_2,X_3$ be the value of the three rolls, and $X = \min\{X_1,X_2,X_3\}$. Note that $X\geq t \iff X_1 \geq t, X_2 \geq t, X_3 \geq t$. By independence, we thus have $$E[X] = \sum_{k=0}^{100} P(X_1 \geq k)P(X_2 \geq k)P(X_3 \geq k) = \frac{1}{100^3}\sum_{k=0}^{100}(100-k)^3 = \frac{1^3 + \cdots + 100^3}{100^3},$$ which simplifies to $25$, which is what your friend claimed.

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