Let some events be modeled by the Poisson process. Find the probability that no more than three events occurred by the time 3t, while at least 1 event occurred by the time t
I've tried to calculate $Pr(N_{3t} \le 3 |N_{t} \ge 1) = \frac{Pr(N_{3t} \le 3 \bigcap N_{t} \ge 1)}{Pr(N_{t} \ge 1)}$.
So $Pr(N_{t} \ge 1) = 1 - Pr(N_{t} = 0) = 1 - exp^{-\lambda t}$.
Then I tried to find $Pr(N_{3t} \le 3 |N_{t} \ge 1)$ using the equation $(S_n > t) = (N_t < n)$.
Am I doing something wrong?
I would consider the intervals $I_1=[0,t]$ and $I_2=[0,2t]$ and calculate
$P(N_t=1)P(0\leq N_{2t}\leq 2) + P(N_t=2)P(0\leq N_{2t}\leq 1)+ P(N_t=3)P(N_{2t}=0)$