# Poisson Process Arrival Rate

I have the following question; Assuming there are a stream of people entering a shop at a Poisson process of rate 5 per second. The arrival requests are in 3 categories:

Girls with probability 0.5 Ladies with probability 0.3 Women with probability 0.2

• In order to know the probability that any given request will be followed by another one within half a second will be ?

So here what i think the result will be is : T~Exponential(5) so P(T<0.5"half a second") = 0.5 -exp^-5 => 0.4932.

• But how can i know, the mean arrival time between two successful women arrivals ?

• And also how can i know the probability that there are 6 ladies arriving in a 10 second period ?
• I would love to know the difference between ladies and women – Henry Nov 17 '14 at 11:45
• What have you tried? If you tell us this then we will be better able to help you. And it helps us feel that we are not just doing your homework for you. – user1729 Nov 17 '14 at 12:41

• Overall $5$ females per second, so $2.5$ per half-second, and the probability of no females in a half-second is $\exp(-2.5)$
• Women arrive at a rate of $0.2 \times 5=1$ per second, so the expected time between women is $\frac11=1$ second
• Ladies arrive at a rate of $0.3 \times 5=1.5$ per second, so $15$ per ten seconds, the the probability of $6$ per ten seconds is, from the Poisson distribution, $\exp(-15)\frac{15^6}{6!}$