I'll take a shot at this one. Binomial covers every situation, Poisson is an 'easier' approximation that is useful in certain situations. Because Binomial contains the n! term, if n > 69 (n is sample size) many calculators run out of steam trying to calculate such a huge number. Poisson gets around this by not using n!, just k!.
Poisson is appropriate when the sample is very large and the chance of any defect is small. You can prove this to yourself using Excel. Create 3 graphs, one using the binomial with n=200 and p=0.03, another using n=20 and p=0.3, and the third using Poisson np=6. (note that all three of these graphs have the same expected value, 6) If you plot all three on the same axis you will see that the B(200,0.03) curve and the Poisson curve are very similar, but the B(20, 0.3) curve is markedly different.
So which is correct? Binomial is always correct, these graphs just show that Poisson 'approximates' the binomial best when the conditions of large sample and small p are met.
Another useful application for Poisson is when np is known, but not n nor p. For example, a contractor averages 7 new homes every year (np = 7). What is the number of people that could call the contractor
? It's huge, but we don't know what it is. What is the probability that any one person in the population that might call wants a house built that year(p)? We don't know, but probably small. But we do know np=7, so we could calculate the probability of the contractor building 6 houses next year, or 8 houses next year.
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