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I am writing a program and am confused with deciding on whether to use a Binomial or Poisson distribution. I will give an example of an identical problem.

Let's say I have a deck of 52 cards, and our random variable X = the # of times (with replacement of course) I choose a King of Hearts.

Let's also say that each day I take note of how many times I managed to draw a King of Hearts (lets just assume that I recorded the occurrences for 3 hours each day).

If I were to choose the Poisson distribution, assume that after 3 days I summed a total of 30 occurrences (1st day = 10, 2nd day = 5, 3rd day = 15) where $\lambda$ = 30/3 = 10 occurrences per day. Now I want to find P(X=1) or the probability of choosing a single King of Hearts on any given day.

Given this context, would it make sense to use this distribution to compute for P(X = 1)? With the way I have explained it, it seems it would be correct but I am not sure.

Please note: I am by no means an expert when it comes to statistics concepts, so I may have not been able to explain this the best; if you are confused or if there exists an error in the logic provided please let me know.

Kemal Zebari
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  • Maybe see here: https://math.stackexchange.com/questions/1050184/difference-between-poisson-and-binomial-distributions. In general, if $n p_n \to \lambda$ as $n \to \infty$, then the distribution $\text{Bin}(n,p_n)$ converges to a Poisson distribution with mean $\lambda$ as $n \to \infty$. Since the probability of drawing a King of Hearts is relatively low ($1/52$) and your number of draws is relatively high, the Poisson approximation should be reasonable. –  Dec 29 '20 at 09:33
  • @Furrer Alright, this makes much more sense now, thanks for your help (and apologies for the repeated comment mentions as I had another question but was able to work it out). – Kemal Zebari Dec 29 '20 at 18:55
  • No problem. I'll submit my comment as an answer; maybe you could accept it for completeness? –  Dec 29 '20 at 19:05

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Maybe see here: Difference between Poisson and Binomial distributions.. In general, if $n p_n \to \lambda$ as $n \to\infty$, then the distribution $\text{Bin}(n,p_n)$ converges to a Poisson distribution with mean $\lambda$ as $n\to\infty$. Since the probability of drawing a King of Hearts is relatively low ($1/52$) and your number of draws is relatively high, the Poisson approximation should be reasonable.