Question

Records show that the probability is 0.000002 that a bike rider will have an accident while driving on the way of university through Biswamil. Using poisson distribution determine the probability that among 200,000 bike riders driven on the way of university through Biswamil, not more than one will have an accident. ( e^-0.4 = 0.67)                                                                                     

Answer 1

For those situations in which $n$ is large and $p$ is very small, the Poisson distribution can be used to approximate the binomial distribution. 

$$\mu=np=\lambda$$

$$\lambda=0.000002(200000)=0.4$$

$$P(X\leq1)=P(X=0)+P(X=1)$$

$$=\dfrac{e^{-0.4}\cdot0.4^0}{0!}+\dfrac{e^{-0.4}\cdot0.4^1}{1!}=1.4e^{-0.4}$$

$$\approx1.4(0.67)=0.938$$