Question

Men arrive at a health centre and at random, at a constant rate of 0.2 per minute. Women
arrive at the same health centre independently and at random, at a constant rate of 0.3 per
minute.
(i) Find the probability that at least three women arrive at the health centre during a 5-
minute period. [4]
(ii) Find the probability that at least 2 men and at least 3 women arrive at the health
centre during a 5-minute period. [5]

Answer 1

i) Let random variale $\xi$ is the number of women arrived at the health centre during a 5-minute period. Then $\xi\in$ Poisson ($\lambda$) where $\lambda=5(0.3)=1.5$.

We will find $P\{\xi=0\}, P\{\xi=1\}, P\{\xi=2\}$ and calculate sum of these probabilities.

Using cumulative table we get $0.8088.$ Then our probability is $1-0.8088=0.1912.$

ii) Let random variable $\eta$ is the number of men arrived at the health centre during a 5-minute period. Then $\eta\in$ Poisson ($\lambda$) where $\lambda=5(0.2)=1.$

We will find $P\{\xi=0\}, P\{\xi=1\}$ and calculate sum of these probabilities.

Using cumulative table we get $0.7358$. Then probability that at least 2 men arrive at the health centre during a 5-minute period is $1-0.7358=0.2642.$

Then our probability is $(0.2642)(0.1912)\approx 0.05$ (events are independent).