Given x1, x2,...,xn is iid over a uniform [0,1], what is the probability that x1+x2+...+xn < 1?
Every explanation I've found seems to use simplex/n-dimension geometry which I can't understand. What's a probability/intuitive way of understand the result of $\frac{1}{n!}$?
My questionable logic is that 1/2 + 1/4 + 1/8 ... = 1 towards infinity. So for the sum of finite n < 1, the probability has to use that logic. So $\sum \frac{1}{2^i}$, then since order doesn't matter, multiply it by n to get $n\sum \frac{1}{2^i}$.
Now out of the (probably) many issues, the most first is that I can't get the probability of any point in a continuous distribution. That first 1/2 is based off of the mean.
