Question: A restaurant gets an average of 5.3 complaints per day from its patrons. Use the Poisson distribution formula to find the probability that on a given day the restaurant will receive  exactly 4 complaints

Answer:

[tex]P(X=4) = 0.164125[/tex]

Step-by-step explanation:

Given

[tex]\bar{x} = 5.3[/tex] — average

Required

Determine the probability of receiving 4 complaints in a day

Since, it follows a poisson distribution, the probability is calculated as:

[tex]P(X=x) = \frac{\bar{x}^x * e^{-\bar{x}}}{x!}[/tex]

In this case:

[tex]x = 4[/tex]

So, we have:

[tex]P(X=4) = \frac{5.3^4 * e^{-5.3}}{4!}[/tex]

[tex]P(X=4) = \frac{789.0481 * 0.004992}{4*3*2*1}[/tex]

[tex]P(X=4) = \frac{3.939}{24}[/tex]

[tex]P(X=4) = 0.164125[/tex]

Hence, the probability of exactly 4 complaints is 0.164125
.