Answer:
The probability is 0.2650
Step-by-step explanation:
Let's start assuming that men and women come in at the same rate.
Let's define the following random variables :
X : ''Number of people that enter a drugstore''
M : ''Number of men that enter a drugstore''
W : ''Number of women that enter a drugstore''
The number of people will be the number of men plus the number of women
⇒
X = M + W
We are also assuming that M and W are independent random variables.
X ~ Po (10)
M ~ Po (λ1)
W ~ Po (λ2)
λ1 = λ2 because we assumed that men and women come in at the same rate.
λ1 = λ2 = λ
λ1 + λ2 = λ + λ ⇒ 2λ = 10 ⇒ λ = 5
M ~ Po (5)
W ~ Po (5)
Because X is the sum of two independent Poisson random variables.
We are looking for :
[tex]P(M\leq 3/W=10)=P(M\leq 3)[/tex]
Because we assume independence.
[tex]P(M\leq 3)= P(M=0)+P(M=1)+P(M=2)+P(M=3)[/tex]
[tex]P(M=m)=\frac{5^{m}}{m!}e^{-5}[/tex] because is a Poisson random variable with λ = 5
[tex]P(M\leq 3)=\frac{5^{0}}{0!}e^{-5}+\frac{5^{1}}{1!}e^{-5}+\frac{5^{2}}{2!}e^{-5}}+\frac{5^{3}}{3!}e^{-5}[/tex]
[tex]P(M\leq 3)=e^{-5}+5e^{-5} +(\frac{25}{2})e^{-5} +\frac{125}{6}e^{-5}=0.2650[/tex]
[tex]P(M\leq 3)=0.2650[/tex]
