Cartridge Walk

Poisson Probability???

An office has 10 printers that each (randomly) reqire new ink cartridges about once per month. If 2 ink cartridges are on hand, what is the chance that there will be enough cartridges to make it through the month?

I know I should use the Poisson Distribution function:
P(x) = (x^λ)(e^λ)/x!

but I'm not sure how to find λ.

Anybody who could walk me through how to do do this problem?? Thanks!

The Poisson distribution can be derived from the binomial distribution. The Poisson is nothing more than the limiting case of the Binomial where n is large and p is small.

A good way to identify when you need to use the Poisson distribution is when the problem requires you to use a rate. This is not always true, but more often than not remembering this will help you to identify a Poisson model.

Let X be the number of printers needing a new print cartridge. X has the Poisson distribution with parameter λt = 10

In general you have:

X ~ Poisson( λt )
P(X = x) = ( λt )^x * exp( -λt ) / x! for x = 0, 1, 2, 3, 4, ...
P(X = x) = 0 otherwise

the mean of the Poisson distribution is the parameter, λt
the variance of the Poisson distribution is the parameter, λt

In this problem we have
λ = 1
t = 10 time unit(s) {printers in this case}

this results in our random variable X ~ Poisson( 10 )

The probability that there will be enough ink cartridges on hand is:

Find P(X ≤ 2 ) =

2
∑ P(X = i)
i=0

= 0.002769396