Poisson or Binomial distribution?

When to use the Poisson or Binomial distribution?:

"If a mean or average probability of an event happening per unit time/per page/per mile cycled etc., is given, and you are asked to calculate a probability of n events happening in a given time/number of pages/number of miles cycled, then the Poisson Distribution is used.

If, on the other hand, an exact probability of an event happening is given, or implied, in the question, and you are asked to calculate the probability of this event happening k times out of n, then the Binomial Distribution must be used."

(Adapted from this page).  Therefore, if there is on average 2 bank failures per month, what is the probability that there are no bank failures in a month?
Poission Distribution (lambda t) = (2 errors per page * 1 page) = 2.
Hence P0 = 2^0/0! * exp(-2) = 0.135

There are 20 banks in a state, the probability of one going bust is 0.1.  What is the probability of losing two banks?

Here it is binomial with n = 20.   Expand (q + p)^20
q^20 + 20 q^19 p + 20(20-1)/2! q^18 P^2  + ...

So P(2) = 20(20-1)/2! q^18 p^2 = 0.285.