Please help with the following statistics problem about Poisson distribution.?

Question

If a drop of water is placed on a slide and examined under a microscope, the number of x of a particular type of bacteria present has been found to have a Poisson probability distribution. Suppose the maximum permissible count per water specimen for this type of bacteria is five. If the mean count for your water supply is two and you test a single specimen, is it likely that the count will exceed the maximum permissible count? Explain.

Answer 1

The poisson distribution of a random variable X (with parameter t) has a probability density function given by (e^-t)(t^x)/x!

So if X - number of bacteria present in your sample, what we want to find is P[X>5]

But to do so, we have to find t. We do know however that t is just the mean of our random variable. Thus t=2.

P[X>5]
=1-P[X=1,2,3,4,5]
=1-[(e^-2)(2)+(e^-2)(4)/2+(e^-2)(8)/6+(e^-2)(16)/24+(e^-2)(32)/120]
=1-(e^-2)[2+2+(4/3)+(2/3)+(4/15)]
=appox. 0.151899

whether this percentage (15+%) is a likely one or not depends on interpretation.