Answer to Question #215873 in Statistics and Probability for Sampson Nsiah

Question #215873

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 


1
Expert's answer
2021-07-13T05:06:52-0400

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

Given λ=3.\lambda=3.


a)

P(X3)=P(X=0)+P(X=1)+P(X=2)P(X\leq 3)=P(X=0)+P(X=1)+P(X=2)

+P(X=3)=eλλ00!+eλλ11!+P(X=3)=\dfrac{e^{-\lambda}\cdot\lambda^0}{0!}+\dfrac{e^{-\lambda}\cdot\lambda^1}{1!}

+eλλ22!+eλλ33!+\dfrac{e^{-\lambda}\cdot\lambda^2}{2!}+\dfrac{e^{-\lambda}\cdot\lambda^3}{3!}

=e3(6+18+27+27)6=13e30.647232=\dfrac{e^{-3}(6+18+27+27)}{6}=13e^{-3}\approx0.647232


b)

P(X3)=1P(X=0)P(X=1)P(X=2)P(X\geq 3)=1-P(X=0)-P(X=1)-P(X=2)

=1eλλ00!eλλ11!eλλ22!=1-\dfrac{e^{-\lambda}\cdot\lambda^0}{0!}-\dfrac{e^{-\lambda}\cdot\lambda^1}{1!}-\dfrac{e^{-\lambda}\cdot\lambda^2}{2!}

=1e3(2+6+9)2=18.5e30.576810=1-\dfrac{e^{-3}(2+6+9)}{2}=1-8.5e^{-3}\approx0.576810




c)

P(X5)=1P(X=0)P(X=1)P(X=2)P(X\geq 5)=1-P(X=0)-P(X=1)-P(X=2)

P(X=3)P(X=4)=1eλλ00!-P(X=3)-P(X=4)=1-\dfrac{e^{-\lambda}\cdot\lambda^0}{0!}

eλλ11!eλλ22!eλλ33!eλλ44!-\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!}

=1e3(24+72+108+108+81)24=1-\dfrac{e^{-3}(24+72+108+108+81)}{24}


=11318e30.184737=1-\dfrac{131}{8}e^{-3}\approx0.184737


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