Question

Q1

A mine hauls, on average, three dump truck of waste per hour to a waste dump. For a given hour, find the probability that it will haul the following number of truck loads:

a)     At most 3 trucks.

b)     At least 3 trucks.

c)      Five or more 

Answer 1

Let $X=$ the number of trucks per hour: $X\sim Po(\lambda).$

Given $\lambda=3.$

a)

$$P(X\leq 3)=P(X=0)+P(X=1)+P(X=2)$$

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

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

$$=\dfrac{e^{-3}(6+18+27+27)}{6}=13e^{-3}\approx0.647232$$

b)

$$P(X\geq 3)=1-P(X=0)-P(X=1)-P(X=2)$$

$$=1-\dfrac{e^{-\lambda}\cdot\lambda^0}{0!}-\dfrac{e^{-\lambda}\cdot\lambda^1}{1!}-\dfrac{e^{-\lambda}\cdot\lambda^2}{2!}$$

$$=1-\dfrac{e^{-3}(2+6+9)}{2}=1-8.5e^{-3}\approx0.576810$$

c)

$$P(X\geq 5)=1-P(X=0)-P(X=1)-P(X=2)$$

$$-P(X=3)-P(X=4)=1-\dfrac{e^{-\lambda}\cdot\lambda^0}{0!}$$

$$-\dfrac{e^{-\lambda}\cdot\lambda^1}{1!}-\dfrac{e^{-\lambda}\cdot\lambda^2}{2!}-\dfrac{e^{-\lambda}\cdot\lambda^3}{3!}-\dfrac{e^{-\lambda}\cdot\lambda^4}{4!}$$

$$=1-\dfrac{e^{-3}(24+72+108+108+81)}{24}$$

$$=1-\dfrac{131}{8}e^{-3}\approx0.184737$$