I am looking for the origin of the following problem. Let there be three independent fair dice. What is the expectation of the number of tosses until each die has shown six at least once?
One can solve the problem for $n$ independent fair dice. Let $(W_i)_{1\leq n}$ be geometric variables, independent and each with parameter $p=1/6$ (they model the dice). Let $T_n$ be the first time when all $n$ dice have shown a six at least once. Then $$E(T_n)=E\left(\max_{1\leq i\leq n}(W_i)\right)=\sum_{k=1}^n\binom{n}{k}\frac{(-1)^{k-1}}{1-(1-p)^k}$$ for $p=1/6$.
Does anyone know what the origin of this problem is? I thought it was attributed to de Moivre, who did know about the inclusion exclusion principle (that is used here), but i can't find any references.