Answer to Question #199421 in Statistics and Probability for sheharyar

Question

If on the average, (r + 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

Poission distribution function

$$P(X=μ) =\frac{ λ^μ e^-λ}{μ!}\\ μ = number of occurrences \\ λ =mean$$

Given

λ =mean = r+1

(i): 4 or more cars will enter the lot?

λ =mean = r+1

put values in formula

$$here\space we\space will\space find\space for\space 4 \space or\space more\space cars\space then\space formula\space as\\ P(X≥4)=1-P(X<4)\\=1-[P(X=0)+P(X=1)+P(X=2)+P(X=3)]\\ =1-\sum_{0≤μ≤3}\frac{ (r+1)^μ e^{-(r+1)}}{μ!}\\$$

(ii): exactly 4 cars will enter? 

μ = number of occurrences =4

λ =mean = r+1

put values in formula

$$P(X=4) =\frac{ (r+1)^4 e^{-(r+1)}}{4!}\\$$