Question

Men arrive at a clinic independently and at random, at a constant mean rate of 0.2 per minute. Women

arrive at the same clinic independently and at random, at a constant mean rate of 0.3 per minute.

(i) Find the probability that at least 2 men and at least 3 women arrive at the clinic during a 5-minute

period. [4]

(ii) Find the probability that fewer than 36 people arrive at the clinic during a 1-hour period.

Answer 1

(i) Expected number of women in 5 minute $=0.3\times 5 =1.5$

Expected number of men in 5 minute $=5\times 0.2 =1$

$P(\text{ at least 3 women}) =P(X>=3)=1-P(X<=2)$

$=1-(\dfrac{e^{-1.5}1.5^0}{0!}+\dfrac{e^{-1.5}1.5^1}{1!}+\dfrac{e^{-1.5}1.5^2}{2!})=1-0.80847=0.191153$

$P(\text{ at least 2 men) } =1-P(X<=1)=1-(\dfrac{e^{-1}1^0}{0!}+\dfrac{e^{-1}1^1}{1!})= 1-0.7358=0.264241$

P(at least 2 men and 3 women) $=0.191153\times 0.264241=0.0505$

(ii) Mean arrive of people i.e men and woman $\lambda= \dfrac{0.3 +0.2}{2}={0.5}{2}=0.25$

Expected number of people in 1 hour i.e. 60 minute $=60\times 0.25 =15$

Probability that fewer than 36 people arrive at the clinic during a 1-hour period-

$P(X<36)=\sum _{r=0}^{36} \dfrac{e^{-\lambda}\lambda^r}{r!}$