Finding the expected number of trials in an experiment. Given a uniform probability distribution over $[0, 1]$, a number is randomly selected from this distribution. We have to find the expected number of trials such that the sum of the picked numbers $ >= $ 1.
I have been told that the answer to this question is $ e $ but I'm not sure how to solve this.

Let $E(x)$ be the expected number of trials for reaching a sum of $\ge1$ starting from a sum of $x$. Then
$$
E(x)=1+\int_x^1E(t)\mathrm dt\;.
$$
Differentiating with respect to $x$ yields
$$
E'(x)=-E(x)\;,
$$
with the solutions $E(x)=c\mathrm e^{-x}$, and the condition $E(1)=1$ yields $c=\mathrm e$, so the solution is $E(x)=\mathrm e\cdot\mathrm e^{-x}=\mathrm e^{1-x}$, and the value $E(0)=\mathrm e$ is the desired probability.
