The birthday paradox: tested in real life?

The birthday paradox suggests (mathematically) that (excluding leap years, etc.) in a room of 23 random people, the chances are 50% that two of them will share a birthday (day, not year).
And in a room of 30 random people, the chances go up to 71%.

Now, I've been in a lot of random groups and so far as I can remember, not one had two people who shared a birthday.

I can only find means to do the experiment (DUH)... but I was wondering if anyone knows any results from experiments which prove the theory.

Cheers,

Originally posted by shavixmir
The birthday paradox suggests (mathematically) that (excluding leap years, etc.) in a room of 23 random people, the chances are 50% that two of them will share a birthday (day, not year).
And in a room of 30 random people, the chances go up to 71%.

Now, I've been in a lot of random groups and so far as I can remember, not one had two people who shared ...[text shortened]... I was wondering if anyone knows any results from experiments which prove the theory.

Cheers,

Firstly, it is not a paradox nor is it a 'suggestion'. It is a mathematical fact.

I would like to know how you did your tests in random groups of people. Did you really have a group of 23 people and asked all of them for their birth dates?

I do recall that in school there were people who shared a birthday in our class (of about 40).

Originally posted by twhitehead
Firstly, it is not a paradox nor is it a 'suggestion'. It is a mathematical fact.

I would like to know how you did your tests in random groups of people. Did you really have a group of 23 people and asked all of them for their birth dates?

I do recall that in school there were people who shared a birthday in our class (of about 40).

Well, it was called a paradox on the internet... that's where I read it yesterday.

I didn't do any tests, I'm just thinking back...
Now, I understand that it's a mathematical fact... I just can't remember it actually happening in reality.

So, I was wondering if their are any statistics where someone has actually done practical measurements to see if it's correct (or... in other words... if mathematical fact = practical reality).

An example of a discrepancy would be: something drops a whole distance, but before it drops a whole distance it has to drop half that distance first. Hence something that's dropped, mathematically (I don't know the equation though), can never hit the floor.

And that's why I'm wondering if this isn't something along those lines.

Originally posted by shavixmir
Well, it was called a paradox on the internet... that's where I read it yesterday.

I didn't do any tests, I'm just thinking back...
Now, I understand that it's a mathematical fact... I just can't remember it actually happening in reality.

So, I was wondering if their are any statistics where someone has actually done practical measurements to see if the floor.

And that's why I'm wondering if this isn't something along those lines.

If we say there is X% chance of something S happening during C type of occasion where 0 < X < 100 then there must be a none-zero chance of S not happening ( as well as a none-zero chance of S happening ) during any C type of occasion that you observe and that would not involve a paradox.

So, I was wondering if their are any statistics where someone has actually done practical measurements to see if it's correct (or... in other words... if mathematical fact = practical reality).

The results of any such measurement or test would be inconclusive or even meaningless because, according to the maths, there would be a none-zero chance of S not happening during any finite set of C type of occasion that are observe so that, even if S happening was never observed in such a test in any observed occasion, that would not contradict the maths. Sometimes, according to the maths of probability, unlikely things do happen you know!

An example of a discrepancy would be: something drops a whole distance, but before it drops a whole distance it has to drop half that distance first. Hence something that's dropped, mathematically (I don't know the equation though), can never hit the floor.

That isn't a discrepancy.
If something moves 1/2 way from point A to point B in time T1 and then moves 1/4 distance between point A to point B in time T2 and then moves 1/8 distance between point A to point B in time T3 etc,
then if you add up all those time periods i.e. T1 + T2 + T3 + ….etc , then even though that is an infinite series of periods of time, they can add up to a finite amount of time rather than an infinite amount of time ( thus something dropping will hit the floor! And in finite time ) . It may be slightly anti-intuitive but where people get confused here into thinking there is a paradox here when there isn't is where they lack understanding that an infinite series of non-zero finite quantities ( where periods of time is just one example ) can add up to a finite rather than an infinite quantity! The above is an example of this where
T*½ + T*¼ + T*1/8 + T*1/16 + ... + T*1/infinity = T ( where T is finite and non-zero )
and there is no mathematical discrepancy here.

This is the best explanation I've ever come across, and you're provided with a way of running the test over and over again very quickly:

http://betterexplained.com/articles/understanding-the-birthday-paradox/

As they say, it's hard to get our linear brains around. What I found really interesting while running that test 100 times was that, although up to 3 pairs of birthdays appeared, not once did a triple show up.

Originally posted by Kewpie
This is the best explanation I've ever come across, and you're provided with a way of running the test over and over again very quickly:

http://betterexplained.com/articles/understanding-the-birthday-paradox/

As they say, it's hard to get our linear brains around. What I found really interesting while running that test 100 times was that, although up to 3 pairs of birthdays appeared, not once did a triple show up.

I find the link strange because it speaks of this being a “paradox” when I see none ( nor anything like or even slightly reminiscent of a 'paradox' ) and it being “counter-intuitive” when I find absolutely nothing counter-intuitive about it!
If most people DO find this counter-intuitive, perhaps I just have better intuition than most people? Or perhaps I do not get so easily confused as most people?
Or perhaps I am just strange for not getting confused!?
Am I the only one?

I've really lost it now.

Sweet Jesus. I'm sticking to Middle-Eastern politics...that I understand.

Originally posted by shavixmir
I've really lost it now.

Sweet Jesus. I'm sticking to Middle-Eastern politics...that I understand.

Understanding middle east politics doesn't seem to help the Palestinians or Jews very much. I'm sure they understand perfectly. The part they screw up is called 'obstinance'.

Originally posted by shavixmir
Well, it was called a paradox on the internet... that's where I read it yesterday.

Well thats a really bad label for it. All it is is a higher percentage than people would typically guess.

I didn't do any tests, I'm just thinking back...
Now, I understand that it's a mathematical fact... I just can't remember it actually happening in reality.
Of course you don't remember it because you have never tested it. The fact is that when you are in a group of 23 or more people you do not go around asking them what their birthdays are and whether any two are identical.
But if you can get hold of class registers or employee records or something like that I am sure you could test it.
I have access to passenger records for some flights including birth dates. I probably could test it on that data but it will take a little coding to get it right. I'm not by my development environment right now but will try and remember to do it next time I am.

Originally posted by shavixmir
The birthday paradox suggests (mathematically) that (excluding leap years, etc.) in a room of 23 random people, the chances are 50% that two of them will share a birthday (day, not year).
And in a room of 30 random people, the chances go up to 71%.

Now, I've been in a lot of random groups and so far as I can remember, not one had two people who shared ...[text shortened]... I was wondering if anyone knows any results from experiments which prove the theory.

Cheers,

You don't need to do any "experiment" to confirm the theory - it only assumes that the birth dates are randomly distributed (a reasonable approximation, which is any case gives a lower bound for the chance if I'm not mistaken). But if the experiment were to be done, it would confirm the statistics. You can prove the result quite easily using elementary statistics, I'm sure you can find the derivation somewhere through Google.

Originally posted by humy
I find the link strange because it speaks of this being a “paradox” when I see none ( nor anything like or even slightly reminiscent of a 'paradox' ) and it being “counter-intuitive” when I find absolutely nothing counter-intuitive about it!
If most people DO find this counter-intuitive, perhaps I just have better intuition than most people? Or perhaps I do no ...[text shortened]... as most people?
Or perhaps I am just strange for not getting confused!?
Am I the only one?

People in general tend to be very bad at "intuitive" statistics, and it needs some training to suppress your intuition when it comes to statistics.

I think it is counter intuitive but that is what makes some probability results so much fun. Twist it round and ask the probability that 23 people enter a room and each one who enters has a different birthday to those already in the room it seems more intuitive.

I have tested it lots of times with groups of students and can report unsurprisingly that most of the groups do have 2 people sharing a birthday (especially as I tend to do it on groups with 25-30 students). The problem does assume birthdays are equally distributed, in real life the probability is more than 50% as birthdays are not equally distributed throughout the year (in the UK there seem to be a lot of March birthdays for instance due to summer loving and another cluster in September maybe due to Christmas spirit).

On a spreadsheet you can get a list of 23 random numbers between 1 and 365 (leap years will not make much difference but you can try 366 if you are worried) and see if any two are the same. In excel =RANDBETWEEN(1,365) and then find the mode of the group.

OK, here is my analysis of some data I have access to. These are flights on a chartered aircraft.
Column 1, is the number of passengers on the flight.
Column 2, is the number of flights with no double birthdays.
Column 3, is the number of flights with two double birthdays.
Column 4, is the number of flights with three double birthdays.
Column 5, is the number of flights with four double birthdays.

So, for example, for flights with 23 passengers, there were 2 with no double birthdays, and 2 with 1 double birthday and 4 with two pairs of double birthday. So overall there were more double birthdays than predicted by chance, but with a sample size of 8, that's not unexpected.

Qty 0 1 2 3
1 19
2 24
3 30
4 34 2
5 24 2
6 13 1
7 11 1
8 8 1
9 5 1
10 4
11 4 3
12 2
13 4 1
14 2 2
15 4
16 4 1 1
17 3
18 4 2
19 6 6 1
20 4 4
21 2 4 1
22 1 5 3
23 2 2 4
24 3 1 3
25 5 4 1
26 1 2
27 2 4 3
28 1 1 1
29 2 1
30 2 1
31 1
32 1
33 1
35 1 1
40 2

Originally posted by shavixmir
The birthday paradox suggests (mathematically) that (excluding leap years, etc.) in a room of 23 random people, the chances are 50% that two of them will share a birthday (day, not year).
And in a room of 30 random people, the chances go up to 71%.

Now, I've been in a lot of random groups and so far as I can remember, not one had two people who shared ...[text shortened]... I was wondering if anyone knows any results from experiments which prove the theory.

Cheers,

Hmm...

Originally posted by twhitehead
OK, here is my analysis of some data I have access to. These are flights on a chartered aircraft.
Column 1, is the number of passengers on the flight.
Column 2, is the number of flights with no double birthdays.
Column 3, is the number of flights with two double birthdays.
Column 4, is the number of flights with three double birthdays.
Column 5, is ...[text shortened]... 26 1 2
27 2 4 3
28 1 1 1
29 2 1
30 2 1
31 1
32 1
33 1
35 1 1
40 2

Thank you.