Consider the following game. You have a bowl of 10 eggs. With your 10 eggs, you want to compete in an egg-braking contest for Easter. In other words, you take your egg and bang it against your opponents’ egg. For simplicity, let us only consider only one hit on one side with probability of a tie equal to zero. Each of your eggs has a probability of a loss $p_k,\quad k=\overline{1,10}$. This is an averaged probability, it does not change with change in the opposing egg. Let $\xi_i$ be the random variable "the number of eggs you broke with the i-th egg.
Let $$\xi=\min_{i=\overline{1,10}}\xi_i$$ What is the distribution of $\xi$?
My idea is that for each i, $\xi_i$ is geometrically distributed, but what can we tell about $\xi$? Can it be done with the generating functions?
For each $i$, $\xi_i\sim Ge(p_i)\Rightarrow\xi=\min_i\xi_i\sim Ge(1-\prod_i(1-p_i))$. This follows by induction and the equality $$\min_{i=\overline{1,n}}x_i=\min\{\min_{i=\overline{1,n-1}}x_i,x_n\}$$ For the base of the induction look at this answer.