Testwiki:Reference desk/Archives/Mathematics/2020 June 29

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June 29

Confirmation of probability calculation

Am I correct in thinking that the probability of no overlap between four random selections of 5 items from 63 is (58C5)(53C5)(48C5)/[(63C5)^3]? 2A00:23C6:AA08:E500:98F5:73C0:9C7E:A44F (talk) 14:13, 29 June 2020 (UTC)

Experimentally confirmed. On a run of one million trials, 648260 had no overlap, about 0.15σ 0.05σ off from the expected value from the formula, 64837.9. The obvious generalization also appears to hold experimentally.  --Lambiam 15:07, 29 June 2020 (UTC)

That generalization is given by the formula ${\prod }_{k=0}^{s-1}(\genfrac{}{}{0ex}{}{n-ki}{i})/{(\genfrac{}{}{0ex}{}{n}{i})}^s$, in which $s$ stands for the number of selections, $i$ for the number of items in each selection, and $n$ for the number of elements to choose from. A moment of combinatorial reflection has convinced me that this is correct.  --Lambiam 15:30, 29 June 2020 (UTC)

Thanks. This arose from four ex-colleagues each choosing 5 favourites from a set of 63, and my wondering how likely at least one common choice would be. The first approach was to treat it as a "birthday" problem (63 possible days and 20 persons), then I realised the error of that, in that no coincidence could arise in each quintet. A simulation followed, giving the same 6.48% as you report, then finally I convinced myself that the formula I asked about was right. I think you did ten million trials, though.→2A00:23C6:AA08:E500:ED00:AF95:D6C2:CE64 (talk) 16:47, 29 June 2020 (UTC)

No, just one million; I made a mistake in going from the fraction printed out to the numerator of that fraction.  --Lambiam 22:07, 29 June 2020 (UTC)

Another way of writing the result is

$\frac{(n-i){!}^s}{n{!}^{s-1}(n-si)!}.$

More generally if there are m_i items in the ith selection, then the probability of no overlap is

$\frac{\prod (n-m_i)!}{n{!}^{s-1}(n-\sum m_i)!}.$

--RDBury (talk) 19:18, 30 June 2020 (UTC)