Answer to Question #199446 in Statistics and Probability for sheharyar

Question

If on the average, (5 + 1) cars enter a certain parking lot per minute, what is the probability that during any given minute (i): 4 or more cars will enter the lot? (ii): exactly 4 cars will enter? 

Answer 1

On average 6 cars enter a certain parking lot per minute

So according to Poisson Distribution:

$\lambda=6$

$P(X=x)=\dfrac{e^{-\lambda}\lambda^x}{x!}$

So,

(i) 4 or more cars will enter the lot.

$P(X\geq4)=1-[P(X=0)+P(X=1)+P(X=2)+P(X=3)]$

$= 1-[\dfrac{e^{-6}6^0}{0!}+\dfrac{e^{-6}6^1}{1!}+\dfrac{e^{-6}6^2}{2!}+\dfrac{e^{-6}6^3}{3!}]\\\ \\=1-\dfrac{1}{e^6}[1+6+18+36]\\\ \\=1-0.1512\\=0.8488$

(ii)  exactly 4 cars will enter:

$P(X=4)=\dfrac{e^{-6}6^4}{4!}=\dfrac{54}{e^6}=0.1338$