Birthday Problem extension

Jim37 · 2016-01-16 07:57:34

Hi all. The birthday problem at https://www.mathsisfun.com/data/probability-shared-birthday.html is very interesting.
I actually have a group of 23 people and we have 2 pairs of matching birth dates. If the probability is 50% for 1 matching pair, what are the chances of 2? I did statistics a looong time ago but have forgotten most of it so I`d appreciate your help.
Thanks
Jim

bobbym · 2016-01-16 20:56:15

Hi;

2 or more or exactly 2?

samak · 2016-01-17 13:11:09

Hi. Thanks, I should have made that clear. Both, if it's not too difficult please. Otherwise just 2 or more. Thanks.

bobbym · 2016-01-22 20:22:07

Hi;

So far I tentatively have a probability of 11% for exactly 2 pairs.

samak · 2016-01-25 16:47:20

Thanks. I'm intrigued by the word "tentatively". Is this as complex as it seems?

bobbym · 2016-01-25 17:20:15

Yes, there is no literature on it that I can discover.

Fruityloop · 2016-01-26 01:16:05

I'm not absolutely sure if this is correct, but I think this is right.

Relentless · 2016-01-26 05:57:23

Hi Fruityloop,

What does that make the general formula, for n people and exactly x coincidences? Is it this:

Also, can anyone suggest how this might be adjusted to include leap days?

Edit: The formula above does not work at all. What is the generalised version?

Last edited by Relentless (2016-01-26 06:07:27)

bobbym · 2016-01-26 07:19:38

Hi Fruityloop;

That is correct! Thanks. I kept missing the 4.

Hi Relentless;

Relentless · 2016-01-26 07:29:06

Hi bobbym,

I am trying to replace two with one to confirm the results given in the shared birthday link, so that we may now have a way to find the probability of 3 or 4 or 20 matches for more people, but am not getting any correspondence at all. It should be a very simple adjustment, but it is proving quite annoying lol

Last edited by Relentless (2016-01-26 07:31:12)

bobbym · 2016-01-26 07:32:59

What type of matches? Pairs?

Relentless · 2016-01-26 07:39:31

Yes, pairs. If I assume the four came from two squared, and I replace all instances of 2 with one, I get a silly probability of 72.7% for exactly one pair (silly because it is meant to be 50% for at least one).

But hold on... if I assume the 4 came from 2 times the number of matches, I get 36.3%. Which seems just right considering the chance of 1 or 2 is 47%

Last edited by Relentless (2016-01-26 07:41:39)

bobbym · 2016-01-26 07:45:36

Better run some simulations to backup those ansatz.

Relentless · 2016-01-26 08:19:47

Hi,

This last interpretation is not quite correct either. Although it gets very close to the right answer for 23 people, if I extend to 57 people I get probabilities that add up to over 200%.

It is very easy to do simulations of this kind in Excel. I write 1 in cell A1 and =(FACT(23)*COMBIN(365,23-A1)*COMBIN(23-A1,A1))/(365^23*2*A1) in cell B1. Replace 23 with any number of people desired. But this last expression must be corrected

Last edited by Relentless (2016-01-26 08:26:00)

bobbym · 2016-01-26 08:53:50

Which interpretation?

Relentless · 2016-01-26 09:08:14

I am sorry if I am being vague. The interpretation that both of these statements are correct: the 4 in your equation can be replaced by 2 times the number of matching pairs, and each 2 in your equation represents the number of matching pairs.

I have also ruled out the 4 being replaced by 2 plus the number of matching pairs, or the number of matching pairs squared, while holding the other assumption. Perhaps the 2s are the culprit. I think I need to know more about how the formula was created before I proceed.

Last edited by Relentless (2016-01-26 09:10:57)

bobbym · 2016-01-26 12:22:56

There may not be a simple extension on the 2 pair formula.

Fruityloop · 2016-01-26 16:09:54

I think the general formula, for n people and exactly x pairs is...

So for exactly 3 pairs of people out of 23 having the same birthday the probability is 0.01832728.

Last edited by Fruityloop (2016-01-26 16:19:05)

Relentless · 2016-01-26 16:58:26

Hi Fruityloop,

If you believe in this formula (which is definitely the best we have), you will have to explain some discrepancies between its results and those mentioned at the shared birthdays page.

If I sum all of the probabilities from 1 to 11 (for the chance of at least one match), I get 49.4592%, which is close enough to the expected 50%.
If I sum all of the probabilities from 1 to 28 for 57 people, I get 80.9279%, which is not even close to the expected 99%.
If I sum all of the probabilities from 1 to 15 for 30 people, I get 67.7786%, which differs somewhat from the expected 70.6%.

Also, If I extend to include 0 pairs, I get the answers I would expect to from that page.
I think the formula is correct, I am just leaving something out in the sums

Fruityloop · 2016-01-26 17:47:22

Hi Relentless,
The chances of at least one match contain other situations where there are more than just pairs of people matching birthdays. Three people sharing the same birthday for example. Maybe that accounts for the discrepancy?

Relentless · 2016-01-26 17:58:28

Hi,
Thanks for your help. That must be it. But then there are extra layers to the problem.
We are kind of tumbling down the rabbit hole here xD

samak · 2016-01-28 14:39:45

Well, that seems to have put the cat among the pigeons. Now I don't feel so bad that I had no idea how to do it!

bobbym · 2016-01-28 15:09:33

What is the new question?

Relentless · 2016-01-28 20:27:59

Hi bobbym,

We have a formula for the probability of x matching pairs among n people: it is the one Fruityloop provided. The new question, if anyone dares to pursue it, is what is the formula for x triples (three way shares) among n people? How about quadruples? Quintuples? And in general, can all possibilities (adding up to 1) be identified with a relevant formula?

Math Is Fun Forum

#1 2016-01-16 07:57:34

Birthday Problem extension

#2 2016-01-16 20:56:15

Re: Birthday Problem extension

#3 2016-01-17 13:11:09

Re: Birthday Problem extension

#4 2016-01-22 20:22:07

Re: Birthday Problem extension

#5 2016-01-25 16:47:20

Re: Birthday Problem extension

#6 2016-01-25 17:20:15

Re: Birthday Problem extension

#7 2016-01-26 01:16:05

Re: Birthday Problem extension

#8 2016-01-26 05:57:23

Re: Birthday Problem extension

#9 2016-01-26 07:19:38

Re: Birthday Problem extension

#10 2016-01-26 07:29:06

Re: Birthday Problem extension

#11 2016-01-26 07:32:59

Re: Birthday Problem extension

#12 2016-01-26 07:39:31

Re: Birthday Problem extension

#13 2016-01-26 07:45:36

Re: Birthday Problem extension

#14 2016-01-26 08:19:47

Re: Birthday Problem extension

#15 2016-01-26 08:53:50

Re: Birthday Problem extension

#16 2016-01-26 09:08:14

Re: Birthday Problem extension

#17 2016-01-26 12:22:56

Re: Birthday Problem extension

#18 2016-01-26 16:09:54

Re: Birthday Problem extension

#19 2016-01-26 16:58:26

Re: Birthday Problem extension

#20 2016-01-26 17:47:22

Re: Birthday Problem extension

#21 2016-01-26 17:58:28

Re: Birthday Problem extension

#22 2016-01-28 14:39:45

Re: Birthday Problem extension

#23 2016-01-28 15:09:33

Re: Birthday Problem extension

#24 2016-01-28 20:27:59

Re: Birthday Problem extension

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