Percentiles and Box Plots · Superchallenge problem

Waiting times $X$ (in minutes) at a help desk satisfy $P(X > t) = e^{-t}$ for every $t \ge 0$. The quartiles of this distribution are the times $Q_1$ and $Q_3$ with $P(X \le Q_1) = \tfrac14$ and $P(X \le Q_3) = \tfrac34$, and a time is an outlier if it lies more than $1.5 \cdot \text{IQR}$ below $Q_1$ or above $Q_3$. What fraction of all waiting times are outliers? Give the exact value.