Question

Suppose that an emergency medicine unit registers on the average 3 patients with paracot poisoning per week. Calculate the probability that in a given week the unit will register, (i) one or more cases of paracot poisoning, (ii) two or more cases but less than 5 cases of paracot poisoning

Answer 1

Let $X=$the number of patients with paracot poisoning registered per week: $X\sim Po(\lambda t).$

Given $\lambda t=3$

(i)

$$P(X\geq 1)=1-P(X=0)=1-\dfrac{e^{-3}(3)^0}{0!}$$

$$=1-e^{-3}\approx0.950213$$

(ii)

$$P(2\leq X<5)=P(X=2)+P(X=3)$$

$$+P(X=4)=\dfrac{e^{-3}(3)^2}{2!}+\dfrac{e^{-3}(3)^3}{3!}+\dfrac{e^{-3}(3)^4}{4!}$$

$$=e^{-3}(\dfrac{9}{2}+\dfrac{9}{2}+\dfrac{27}{8})=\dfrac{99e^{-3}}{8}\approx0.616115$$