I'm solving a problem of two independent Poisson processes $N^1_t \text{ and } N^2_t$. An some point I have to compute the probability that the third arrival of the second process be less than the first arrival of the first process i.e.: $$P\left[ S^2_3< S^1_1 \right]$$ Where $S_n$ is the arrival time of the nth event.
What I've done: $$P\left[ S^2_3< S^1_1 \right]= \int_0^\infty P\left[ S^2_3< S^1_1| S^1_1=x\right] P\left[ S^1_1=x\right]dx$$ $$= \int_0^\infty P\left[S^2_3< x\right] P\left[ S^1_1=x\right] dx$$ $$= \int_0^\infty \left(\int_0^{x}P\left[S^2_3= u\right]du \right) P\left[ S^1_1=x\right] dx$$ And due to $S_n \sim gamma(n,\delta)$ the integral could be solved by using the PDF, however it is a long way.
Is there an easy way to compute this probability?