Question

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?

Answer 1

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 `color(blue)k` tankers arriving in a single day
given the average of `color(magenta)(lamda=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
`color(white)("XXX")1-P(<=12)`

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