Let $n> 1$ be an integer and let $x_1,\ldots,x_n$ be positive real numbers, all between $0$ and $1$. Is it possible to prove that $$ \frac{\sum_{i=1}^{n}x_i}{1-\prod_{i=1}^{n}(1-x_i)}\ $$ will increase as $x_1,\ldots,x_n$ increases? ($n$ must stay the same)
This is a part of the proof on probability theory. Numerator is the sum of probabilities and denominator is the probabilities of at least one event, and $x_i$ and $y_i$ are probability units of events.
I thought about using derivative to show that decreasing any $x_i$ will decrease the whole thing, but wasn't sure how to apply derivative on product/sum.
Is calculus a necessity (I know it is in the title, but in the body that just seems to be your method) ?
If not you have that, with $f(n)$ being the expression you give, $$f(n+1)=\frac{\sum_{i=1}^{n}{x_i}+x_{n+1}}{1-(1-x_{n+1})\prod_{i=1}^{n}{(1-x_i)}} $$
It's easy to see that $f(n+1)>f(n)$ as the denominator has decreased $(1-x_{n+1}<1)$ while the numerator has increased, so this function is increasing.