Phone calls arrive at a telephone exchange at an average rate of $1$ hit per minute ($\lambda=1$). Let $T_4$ be the time in minutes when the fourth hit occurs after the counter is switched on.
Find $\mathbb{P}(3\le T_4\le 5)$.
So to do this we calculate $\mathbb{P}(T_4\le 5)-\mathbb{P}(T_4<3)$. I calculate $\mathbb{P}(T_4\le 5)=1-\mathrm{e}^{-5(118/3)}$, using the Poisson process $1-\mathbb{P}(\mathrm{Pois}(5)\le 3)$. But the problem is I'm not sure how to compute $\mathbb{P}(T_4<3)$, namely that the fourth arrives in less than 3 minutes. I know that it will be another Poisson with mean $3$, but I'm confused on the sign of the inequality and number of terms.
Thanks!!