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.

The expected number of trials is the sum over $k=0,1,\ldots$ of the probabilities that the experiment isn't over after $k$ numbers have been drawn. The probability for $k$ numbers to sum to less than $1$ is the volume of the $1/k!$-th part of the $k$-dimensional unit cube, i.e. $1/k!$, so the expected number of trials is $\sum_{k=0}^\infty1/k!=\mathrm e$.
