Question

The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 7. Find the probability that more than four road construction projects are currently taking place in the city

Answer 1

Let $X=$ the number of road construction projects: $X\sim Po(\lambda).$  

Given $\lambda=7.$

$$P(X>4)=1-P(X\le 4)$$

$$=1-P(X=0)-P(X=1)$$

$$-P(X=2)-P(X=3)-P(X=4)$$

$$=1-\dfrac{e^{-\lambda}\cdot\lambda^0}{0!}-\dfrac{e^{-\lambda}\cdot \lambda^1}{1!}$$

$$-\dfrac{e^{-\lambda}\cdot\lambda^2}{2!}-\dfrac{e^{-\lambda}\cdot \lambda^3}{3!}-\dfrac{e^{-\lambda}\cdot \lambda^4}{4!}$$

$$=1-e^ {-7}(1+7+49/2+343/6+2401/24)$$

$$\approx 0.82701$$