Question

Suppose that the probability of suffering a side effect from a certain vaccine is $0.005$. If $1000$ persons are inoculated, find the probability that:
(i) atmost $1$ person suffers
(ii) $4,5$ or $6$ persons suffer $[e^{-5}=0.0067$]

Answer 1

Let this probability of suffering from side effect be $p$.
$n=1000,p=0.005,A=np=5$
(i) $P$ (atmost $1$ person suffer)$=P(x\le 1)$
$=P(x=0)+P(x=1)$
$=\frac{e^{-\lambda }.{\lambda }^0}{0}+\frac{e^{-\lambda }.{\lambda }^1}{1}$
$=e^{-\lambda }(1+\lambda )=e^{-5}(1+5)=6\times 0.0067=0.0402$
(ii) $P$ ($4,5$ or $6$ persons suffer)
$=P(x=4)(x=5)(x=6)$
$=P(x=0)+P(x=1)$
$\frac{e^{-\lambda }.{\lambda }^4}{4}+\frac{e^{-\lambda }.{\lambda }^5}{5}+\frac{e^{-\lambda }.{\lambda }^6}{6}$
$=\frac{e^{-\lambda }.{\lambda }^4}{4}[1+\frac{1}{5}+\frac{{\lambda }^2}{30}]$

$=\frac{e^{-5}.5^4}{24}\left[\frac{17}{6}\right]=0.4944$