Suppose that males and females enter a bank with respect to two independent Poisson processes with intensities $\lambda_{m}=3$ and $\lambda_{f} = 5$ per hour, respectively. Find the probability that a male enters the bank sooner than a female at a certain hour.
I've been told that we're looking for $P(S_{m}<S_{f}) = \frac{\lambda_{m}}{\lambda_{m}+\lambda_{n}}$. What do the terms $S_{m}$ and $S_{n}$ represent? Why does $P(S_{m}<S_{f}) = \frac{\lambda_{m}}{\lambda_{m}+\lambda_{n}}$ hold?
$S_m$ is the time of first arrival in the first process and $S_f$ is the time of first arrival in the second process. $S_m >t$ is the event that there is no arrival up to time $t$ and so $P\{S_m >t\} =e^{-\lambda_m t}$. By differentiation we see that $S_m$ has density $\lambda_m e^{-\lambda_m t}$. Similarly, $P\{S_f >t\} =e^{-\lambda_f t}$ and $S_f$ has density $\lambda_f e^{-\lambda_f t}$. Now, by conditioning on $S_m$ we get $P\{ S_m <S_f\}=E e^{-\lambda_f {S_m}}$. Hence $P\{ S_m <S_f\}=\int_0^{\infty} e^{-\lambda_f t} \lambda_me^{-\lambda_m t}\, dt=\frac {\lambda_m} {\lambda_m+\lambda_f}.$