Question

Use the following information to answer the next six exercises: On average, a clothing store gets$120$ customers per day.

What is the probability that the store will have fewer than $12$customers in the first two hours?

Short answer

The probability of getting smaller than $12$guests in the first two hours at the store i.e.,$P(Y<12)$is $0.0214$

Step-by-step solution

:Given 

Number of guests per day$=120$
Number of opening hours per day $=12$
Let us define a arbitrary variable$Y$ as
$Y=$number of guests arriving in the first two hours at the apparel store.$Y$ follows a Poisson distribution
Formulaused
The probability of$y$successes is the given as
localid="1648544017477" $P(Y=y)=\frac{e^{-\lambda }{\lambda }^y}{y\mathbf{\text{!}}}=;y=0,1,2,3,......\&\lambda >0$

Calculation

Number of guests per hour$=120/12=10$
Number of guests per two hours$=10*2=20$
Thus, in this case$\lambda =$average number of guests per two hours$=20i.e.,$$Y~P\left(20\right).$
We want to find the probability of getting smaller than $12$ guests in the first two hours i.e.,$P(Y<12)$. This can be attained as
$P(Y<12)=P(Y=0)+P(Y=1)+P(Y=2)+.....+P(Y=11)$
$=\sum _{y=0}^{11}\frac{{\displaystyle e^{-\lambda }{\lambda }^y}}{{\displaystyle y\mathbf{\text{!}}}}=0.0214$