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?

1 Answer
Alan P.
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 #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:
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