Answer to Question #128157 in Statistics and Probability for mahi

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.