The probability of the first arrival

Question

In the Poisson process, the arrival rate for man and female is $15$ and $20$ respectively. What is the probability of the first arrival is a man?

I know it is $\frac {15}{35}$, but how to derive this?

Answer 1

Let $M(t)$ and $F(t)$ be Poisson processes with rates $\lambda$ and $\mu$, respectively. Let $T_1 = \inf\{t>0: M(t) = 1\}$ and $S_1 = \inf\{t>0: F(t) = 1\}$. Then $T_1$ and $S_1$ have exponential distribution with parameter $\lambda$ and $\mu$, respectively. It follows that \begin{align} \mathbb P(T_1<S_1) &= \iint_{\mathbb R^2} f_{T_1}(t)f_{S_1}(s)\mathsf 1_{\{t<s\}}\ \mathsf d(t\times s)\\ &= \int_0^\infty \int_0^s \lambda e^{-\lambda t}\mu e^{-\mu s}\ \mathsf dt\ \mathsf ds\\ &= \int_0^\infty \mu \left(e^{\lambda s}-1\right) e^{-s (\lambda +\mu )}\ \mathsf ds\\ &= \frac\lambda{\lambda+\mu}. \end{align} In this case $\lambda=15$ and $\mu=20$, so $$ \frac\lambda{\lambda+\mu} = \frac{15}{15+20} = \frac{15}{35}. $$