Poisson Approximation When "p" is Large

Question

So we use poisson to approximate binomial distributions when "n" (trials) is large and "p" (probability of success) is small.

What about when "p" is large and "q" (probability of failure) is small?

Would it be 1 - P(k successes) where we substitube "q" for "p" in the mean above?

Answer 1

In addition to my Comment, you seem to be looking for some intuition about binomial vs. Poisson models. Here is an example that might help.

Suppose you want a probability model for the number of customers at a small shop on a particular day.

Binomial model. If there are $n = 10,000$ potential customers and the probability any one of them comes to the shop on any one day (independently of each other) is $p = 0.002,$ Then you could model the number of customers on a day as $X \sim \mathsf{Binom}(n=10,000,\,p = 0.002)$. Notice that $\mu_X = np = 20$ and $\sigma_X^2 = np(1-p) = 19.96,$ numerically, nearly the same as $\mu.$

Poisson model. More simply, suppose you have no idea how many potential customers there are, but that past records show an average of 20 customers a day. Then we can model the number of customers on a day as $Y \sim \mathsf{Pois}(\lambda = 20).$ In this case, $\mu_Y = \lambda = 20$ and $\sigma_Y^2 = 20,$ numerically, exactly the same as $\mu.$

Theoretical distinctions: The Poisson model allows for the possibility that the same person may patronize the shop more than once in the same day, while the binomial model does not. Also, the Poisson model puts no theoretical limit on the number of customers in a day, whereas the binomial model does not envision the possibility of more than $n = 10,000.$ (For practical purposes, this is not an important distinction because probabilities for huge numbers of customers are essentially $0$ for the Poisson model.)

Numerical similarities: The following brief session in R statistical software shows probabilities of several numbers of customers in a day according to both models. They are very nearly the same.

n = 10000;  p = .002;  lam = 20;  nr.cust = 15:25
bino = round(dbinom(nr.cust, n, p), 4)
pois = round(dpois(nr.cust, lam), 4)
cbind(nr.cust, bino, pois)
      nr.cust   bino   pois
 [1,]      15 0.0516 0.0516
 [2,]      16 0.0646 0.0646
 [3,]      17 0.0760 0.0760
 [4,]      18 0.0845 0.0844
 [5,]      19 0.0889 0.0888
 [6,]      20 0.0889 0.0888
 [7,]      21 0.0847 0.0846
 [8,]      22 0.0770 0.0769
 [9,]      23 0.0669 0.0669
[10,]      24 0.0558 0.0557
[11,]      25 0.0446 0.0446

The figure below shows binomial probabilities as vertical bars, and Poisson probabilities as open circles. Within the resolution of the plot, the two models are not distinguishable.