I want to calculate $P(S_k < t < S_{k+1})$, where $S_k$ is a sum of interarrival times of the Homogenuous Poisson process. $\{S_{k+1} < t\} \subseteq \{S_{k+1} < t\}$.
I firstly 'cleared' the possibility of $\{S_{k+1} < t\}$:
$$ i)\ P(S_k < t < S_{k+1})= P(S_k < t) - P(S_{k+1}< t ) $$
I am not sure if anything else needs to be added to the equation i) - what bothers me is the fact, that I do not know if I $P(S_{k+1} > t) $ is included in the equation $i)$
I would be thankful if someone would explain to me these basic features of probability and Poisson processes.
Let $N(t) \sim Poiss(\lambda t)$, hence, the inter-arrival times between $k$ arrivals is $S_k \sim \mathcal{E}rlang(k, \lambda)$. This is due to the fact that $S_1 \sim \exp(\lambda)$.
So you can compute \begin{align} P(S_k \le t) &= \int_{0}^t\frac{\lambda^ k}{(k-1)!}e^{-\lambda }x^{k-1}dx \\ &=P(N(t)\ge k)\\ &= \sum_{j=k}^{\infty}e^{-\lambda t}\frac{(\lambda t)^j}{j!}\\ &=1- \sum_{j=0}^{k-1}e^{-\lambda t}\frac{(\lambda t)^j}{j!}. \end{align}
Where the equality $P(S_k \le t) = P(N(t)\ge k)$ stems from the fact that "$k$-th arrival was before time $t$ " is similar to "in $[0,t]$ where at least $k$ arrivals".
I'm not sure that it answers directly your question, but I hope it helps.