# The mean number of oil tankers at a port city is 8 per day. The port has facilities to handle up to 12 oil tankers in day. What is the probability that too many tankers will arrive on a given day?

Aug 28, 2017

Assuming (perhaps without justification) a Poisson distribution,
the probability of more than 12 tankers arriving on a given day is (approximately) color(red)(6.38%)

#### Explanation:

If the tankers arrive with a Poisson distribution (which seems likely but not certain), then
The probability of $\textcolor{b l u e}{k}$ tankers arriving in a single day
given the average of $\textcolor{m a \ge n t a}{l a m \mathrm{da} = 8}$ tankers per day
is given by the formula:
color(white)("XXX")P(color(blue)k)=e^(-color(magenta)lamda) * (color(magenta)lamda^color(blue)k)/(color(blue)k!)=e^(-color(magenta)8) * (color(magenta)8^color(blue)k)/(color(blue)k!)
and
the probability of $12$ or fewer tankers arriving would be
color(white)("XXX")P(<=12)=Sigma_(k=0)^12 e^(-8) * (8^k)/(k!)

The probability of more than $12$ tankers arriving would be
$\textcolor{w h i t e}{\text{XXX}} 1 - P \left(\le 12\right)$

These equations can be evaluated with a calculator or (as below) by using a spreadsheet: 