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 more than $12$customers in the first hour?

Short answer

The probability of getting quite$12$ customers within the first hour at the shop is $0.2084$.

Step-by-step solution

Given

Number of shoppers per day $=120$
Number of opening hours per day $=12$
Let us define a variable quantity $Y$as:
$Y=$ number of shoppers arriving within the first hour at the shop.$Y$ follows a statistical distribution.

Formula Used

The probability of $y$successes is the given as:

$P(Y=y)=\frac{e^{-\lambda }{\lambda }^y}{y!};y=0,1,2,\dots ..\&\lambda >0$

Calculation

Number of customers per hour$=\frac{120}{12}=10$

Therefore, in this case $\lambda =$average number of customers per hour $=10$i.e., $Y~P\left(10\right)$.

We want to find the probability of getting more than 12 customers in the first hour i.e., $P(Y>12)$. This can be obtained as:

$P(Y>12)=1-P(Y\le 12)$

$=1-\left[P\right(Y=0)+P(Y=1)+P(Y=2)+\dots +P(Y=12\left)\right]$

$=1-\sum _{y=0}^{12}\frac{e^{-\lambda }{\lambda }^y}{y!}$

$=1-0.7916=0.2084$