# Statistics help for an old codger **URL:** \ **Category:** Factual Questions\ **Created:** [December 15, 2008, 6:30am UTC](https://boards.straightdope.com/t/statistics-help-for-an-old-codger/477217 "2008-12-15T06:30:13Z")\ **Posts on this page:** 7\ **Page:** 1
**Author:** ![jharvey963](https://avatars.discourse-cdn.com/v4/letter/j/54ee81/32.png) [@jharvey963](https://boards.straightdope.com/u/jharvey963)\ **Post date:** [December 15, 2008, 6:30am UTC](https://boards.straightdope.com/t/statistics-help-for-an-old-codger/477217/1 "2008-12-15T06:30:13Z")
It’s been more decades than I care to admit since I’ve had statistics, and I’d like to ask for some assistance with some counting and odds problems. (For my own edification. These aren’t homework problems.) I remember the formula for “N choose M, without replacement”: N! / (M! (N - M)!) But now, let’s assume that you choose _another_ set without replacement, say P, from the original N. I have some questions about the relationships between M and P. It always helps me to give actual numbers, so let’s say … N is 20 (the original set contains 20 items), M is 2 (2 elements are chosen, without replacement), and P is 5 (5 new elements are chosen, without replacement, from the original 20). What are the formulas for these: 1. How many elements from M would you “expect” to see in P? 2a. What are the odds of finding exactly 0 elements from M in P? 2b. What are the odds of finding exactly 1 element from M in P? 2c. What are the odds of finding exactly 2 elements from M in P? … 2x. What are the odds of finding all elements from M in P? 3a. How big would P have to be for you to expect that P contains exactly 1 element from M? 3b. How big would P have to be for you to expect that P contains exactly 2 elements from M? … 3x. How big would P have to be for you to expect that P contains all elements from M? Thanks, J. ---
**Author:** ![muttrox](https://avatars.discourse-cdn.com/v4/letter/m/a8b319/32.png) [@muttrox](https://boards.straightdope.com/u/muttrox)\ **Post date:** [December 15, 2008, 2:56pm UTC](https://boards.straightdope.com/t/statistics-help-for-an-old-codger/477217/2 "2008-12-15T14:56:47Z")
Silly clarification – I assume you can do the replacing after picking all of M? Otherwise there is no overlap at all. Some basics: M=20 choose 2 = 190 combinations P=20 choose 5 = 15,504 combinations M and P are independent. Question 2c/2x is simply 190/15504 = 1.2% That’s a start - my brain is up to speed on Monday morning yet for the rest. ---
**Author:** ![jharvey963](https://avatars.discourse-cdn.com/v4/letter/j/54ee81/32.png) [@jharvey963](https://boards.straightdope.com/u/jharvey963)\ **Post date:** [December 15, 2008, 5:00pm UTC](https://boards.straightdope.com/t/statistics-help-for-an-old-codger/477217/3 "2008-12-15T17:00:15Z")
> [@muttrox](#): > > Silly clarification – I assume you can do the replacing after picking all of M? Otherwise there is no overlap at all. > … Yes, M elements are chosen first (without replacement). Then those elements are replaced back into N. Then P elements are chosen (without replacement) from the original 20 elements in N. J. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [December 15, 2008, 5:12pm UTC](https://boards.straightdope.com/t/statistics-help-for-an-old-codger/477217/4 "2008-12-15T17:12:17Z")
The number of elements from the first sample in the second sample is distributed [hypergeometrically](http://en.wikipedia.org/wiki/Hypergeometric_distribution) with suitable parameters. ---
**Author:** ![jharvey963](https://avatars.discourse-cdn.com/v4/letter/j/54ee81/32.png) [@jharvey963](https://boards.straightdope.com/u/jharvey963)\ **Post date:** [December 15, 2008, 7:02pm UTC](https://boards.straightdope.com/t/statistics-help-for-an-old-codger/477217/5 "2008-12-15T19:02:17Z")
> [@ultrafilter](#): > > The number of elements from the first sample in the second sample is distributed [hypergeometrically](http://en.wikipedia.org/wiki/Hypergeometric_distribution) with suitable parameters. uhhhhhhh, thanks… The link, while undoubtedly the correct distribution, is pretty dense. Would someone mind giving a more “layman’s” explanation of this, and how it would apply to my problem? Thanks a lot, J. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [December 15, 2008, 7:51pm UTC](https://boards.straightdope.com/t/statistics-help-for-an-old-codger/477217/6 "2008-12-15T19:51:38Z")
I will write “n choose k” as C(n, k). In the simplest version of the hypergeometric distribution, you have a population of N items that can be split into two non-overlapping groups of size M and N - M. You want to sample without replacement from the overall population, and you want to compute the odds of drawing a certain portion of that sample from the first population. If you sample P times, the probability that k of those draws come from the first population is C(M, k)C(N - M, P - k)/C(N, P). The number of elements from the first population you expect to see in your sample is the mean, which is equal to PM/N. For your example, that’s 5\*2/20, or 1/2. You can use that formula to find how many times you have to sample to expect to see a certain number of elements from the first population. It’s instructive to compute how large a sample you have to take to expect to see the entire first population, so I’ll let you work that one out. ---
**Author:** ![nivlac](https://avatars.discourse-cdn.com/v4/letter/n/3bc359/32.png) [@nivlac](https://boards.straightdope.com/u/nivlac)\ **Post date:** [December 15, 2008, 8:06pm UTC](https://boards.straightdope.com/t/statistics-help-for-an-old-codger/477217/7 "2008-12-15T20:06:29Z")
It is indeed the hypergeometric distribution. In the first draw you are essentially selecting m “defective” units out of the N available. Then in your second draw of n units you are looking for these same m units. So N and m are obviously 20 and 2, respectivly, in your example. So what should n and k be in the wiki article? I get 1/19 as the answer.