Question

Close to 15 hospitals in Maharashtra has to be completely or partially shut down after
many of their staff were quarantined following exposure to COVID-19 patients. The
number x of medical staff entering the hospitals treating COVID-19 patients on any
week has a Poisson probability distribution with average equal to 5 staff per week.
(a) What is the probability that the number of staff entering the hospital on a particular
week is two
(b) Is it likely that x will exceed 10? Substantiate your answer.

Answer 1

(a) Poisson probability

$$P(X=x)=\dfrac{e^{-\mu}\cdot \mu^x}{x!}$$

Given $\mu=5$

$$P(X=2)=\dfrac{e^{-5}\cdot 5^2}{2!}\approx0.084224$$

(b)

$$P(X>10)=1-P(X\leq 10)=$$ $$=1-P(X=0)-P(X=1)-P(X=2)-$$ $$-P(X=3)-P(X=4)-P(X=5)-$$ $$-P(X=6)-P(X=7)-P(X=8)-$$ $$-P(X=9)-P(X=10)\approx0.0137$$

We note $P(X>10)=0.0137<0.05.$ We consume that the probability $P(X>10)=0.0137$ is small, which indicates that the event is unlikely to occur by chance and thus it is not likely that $X$ will exceed 10.