Question regarding distributions of min/max functions I am having trouble with the following problem and was wondering if someone can point me in the right direction.
Let $X_1, X_2, \ldots$ be an infinite sequence of independent, identically distributed uniform random variables on $[0,1]$. For a given constant $x$, where $0 < x <1$, define
$$
M= \min\{n\ge1: X_1 + X_2 + \ldots + X_n > x\}.
$$
(So for a given $x$, $M$ is a discrete random variable that tells us out how many terms it takes in order for the above summation to exceed $x$.)
Find $P(M>k)$, where $k$ is a non-negative integer.
I have an inuition that I need to set up some kind of recursion but I am not sure if this approach is correct either.
Thank you for the help.

Clearly $P(M>0)=1$. If you define $S_k = X_1+\ldots+X_k$, then for $k\ge1$, we have $P(M>k) = P(S_k\le x)$, where the formula for $P(S_k\le x)$ can be found in here.
