Answer to Question #225803 in Statistics and Probability for Prathyush

Question

Accidents occur at an intersection at a Poisson rate of 2 per day. What is the probability that
there would be no accidents on a given day ? What is the probability that in January there are
at least 3 days (not necessarily consecutive) without any accidents ?

Answer 1

a) Let $X=$ the number of accidents occured: $X\sim Po(\lambda t)$

$\lambda t=2(1)=2$

$$P(X=0)=\dfrac{e^{-2}\cdot2^0}{0!}=e^{-2}\approx0.13535$$

b) Let $Y=$ the number of days  without any accidents: $Y\sim Bin(n, p)$

$n=31, p=0.13535, q=1-p=0.86465$

$$P(Y\geq3)=1-P(Y=0)-P(Y=1)-P(Y=2)$$

$$=1-\dbinom{31}{0}(0.13535)^0(0.86465)^{31-0}$$

$$-\dbinom{31}{1}(0.13535)^1(0.86465)^{31-1}$$

$$-\dbinom{31}{2}(0.13535)^2(0.86465)^{31-2}$$

$$\approx0.81$$