Earthquakes occur according to a Poisson process with $\lambda = 5$ per year, let $T_1$ be the time of the first occurrence of the process and we wish to figure out what is the average time that will pass until we reach a period of 3 months without any earthquakes.
So, if we define $X$ as the random variable which describes the time until we get a period of 3 months without earthquakes we have that:
\begin{align*} E[X] &= E\left[X|T_1 > \frac{1}{4}\right]\cdot P \left\{ T_1 > \frac{1}{4}\right\} + E\left[X|T_1 \leq \frac{1}{4}\right]\cdot P \left\{ T_1 \leq \frac{1}{4}\right\}\\ &=0 + E\left[X|T_1 \leq \frac{1}{4}\right]\cdot P \left\{ T_1 \leq \frac{1}{4}\right\} \end{align*}
According to my notes, $E\left[X|T_1 \leq \frac{1}{4}\right] = E\left[T_1|T_1 < \frac{1}{4}\right] + E[X]$.
Can someone please explain why the previous equality holds and what is the intuition behind it?