The probability that a visitor to an animal shelter will adopt a dog is .20. out of nine visits, what is the probability that at least one dog will be adopted

This is a binomial probability situation, since a dog either is adopted or is not adopted.  The chances of a dog's being adopted in 0.20.  Here we're speaking of 9 visits.  Thus, n=9, p=0.20.

One way of doing this problem is to calculate the probability that ONE dog will be adopted, and then that that TWO dogs will be adopted, and so on, up to NINE dogs.  Add together these nine probabilities to get your answer.

But a better (faster) approach would be to calculate the probability that ZERO dogs will be adopted, and then to subtract this from 1.000.

Using my TI-84Plus calculator, I figured that P(0 dogs will be adopted) is binompdf(9,0.20,0), or 0.134.  Subtracting this from 1.000, we get 0.866 (answer to this problem).