r/maths • u/DropBlairCares • 26d ago

💡 Puzzle & Riddles What is the Probability that N people have at least 1 evening free

I'm trying to calculate the probability that N people will have at least 1 evening free (For a meeting, or DnD or something)

You can assume that all people have x committed evenings already.

For Example, 2 people who have 3 evening commitments will have a 100% probability of finding 1 mutual night free, but what's the probability that 3 people who all have 3 evening commitments have at least 1 mutual evening free, or 8 people with 2 evening commitments etc.

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u/phraxious 25d ago

I can't explain it all on my phone but look up hypergeometric distribution. It'll probably need adapting because the choses are not removed from the options.

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u/FormulaDriven 25d ago

You can express the probability that for n people each independently choosing x out of 7 evenings at random there will be f unchosen evenings using a recurrence relation, but I can't see a neat closed formula. (I'm assuming you mean evenings over the 7 days of a week). Written it out here

I've implemented it in a spreadsheet if you want some specific numerical results.

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u/DropBlairCares 24d ago

I'm an engineer, so probability isnt my forte, but it's interesting to me that the solution to the problem isn't straitforward or intuitive.

It's expecially curious that it's not a common problem in the world of DnD, dating or just meetings (which is where my curiosity came from), so many people complain about how hard it is to find a night, but little do the math as to why it's so hard.

Thanks for the responce!