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Suppose that in a very large bushel of apples there are 20% rotten ones. What is the chance that a random sample of 10 apple contains 2 rotten ones?

For this question, if the size of the bushel was given, I would solve this problem as if Y was a hypergeometric variable (correct me if I am wrong), but since N (bushel size) is not given, should I treat it as a binomial variable or as a variable that follows a poisson distribution?

Are both valid, why should I choose one and not the other?

For binomial, would I be wrong in doing (10 choose 1)(.20^2)(.80^8)? If this is wrong, is it because I cannot make the assumption that there will be 20% of rotten apples in that sample or could I correctly assume that?

I am not sure about the relationship between binomial and poisson distributed variables, why can you use a poisson distribution to approximate a binomial one? My textbook is very confusing.

Is it as simple as 0.2 * 10 and I'm just going too far with this haha?

Thank you!

nx__
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1 Answers1

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In this case, you are sampling without replacement. This means each time you pick an apple, you are not putting it back into the bushel. So, the hypergeometric distribution would apply. Recall that for the hyper geometric distribution:

$$P(X=x) = \frac{\binom{M}{x}\binom{N-M}{n-x}}{\binom{N}{n}}$$

where:
N = size of bushel
M = 0.2N (since 20% of the apples are rotten)
n = 10 (sample size)
x = number of rotten apples

Now, your question about binomial vs poisson.
Recall, the binomial distribution:

$$P(X=x) =\binom{n}{x}p^x(1-p)^{n-x}$$

If n is very large, the formula is very hard to compute. So, in that case, we would use the Poisson to approximate the binomial.

Recall the Poisson:

$$P(X=x)=\frac{\lambda^x}{x!}e^{-\lambda}$$
where $\lambda=np$

user137481
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  • Thank you! And is it better to either of the two to approximate an hypergeometric variable? I tried both the poisson and the binomial approximations and I get 2 probability values which are rather close. Is either of them more "standard" to approximate a hypergeometric? – nx__ May 05 '14 at 22:12
  • Poisson can be used to approximate the Binomial. I'm not sure if either of them can be used to approximate the hypergeometric. You should pose that question and someone with more expertise than me should be able to answer. – user137481 May 06 '14 at 00:44
  • http://math.stackexchange.com/questions/330553/proof-that-the-hypergeometric-distribution-with-large-n-approaches-the-binomia – user137481 May 06 '14 at 01:39
  • The link provides a proof of how the binomial can be used to approximate the hypergeometric. – user137481 May 06 '14 at 01:40