Chances of 2 people sharing a birthday=50%?

[legion]

I read somewhere once that in a class of 30 children the chances of 2 of the children sharing a birthday are about 50/50.

Is this correct and if it is, why is it?

[One And Only Wanderers]

I find this hard to credit. I have never met anyone who has the same birthday at me, even when I was at University with a much larger class size than 30.

[alterego]

From Wikipedia, "The birthday paradox states that if there are 23 people in a room then there is a slightly more than 50/50 chance that at least two of them will have the same birthday. For 60 or more people, the probability is greater than 99%. This is not a paradox in the sense of it leading to a logical contradiction; it is a paradox in the sense that it is a mathematical truth that contradicts common intuition. Most people estimate that the chance is much lower."

The article goes into the maths involved.

[missbunny]

I thought it was a much smaller number - in a group of 17 people at least two will have the same birthday more than 50% of the time. I think this was discussed here 2-3 years ago.

FYI, it means the same month and day - not the same year. The chances of that happening are much, much lower.

[ultrafilter]

In a class of 23 people, there's a 50% chance that two people have the same birthday. This is a pretty standard demonstration of the fact that probability can be counterintuitive.

However, if you're in a class of 23 people, the probability that somebody shares your birthday is significantly lower than the probability that two people share the same birthday.

[missbunny]

23 people sounds more right - I think I misremembered the 17.

[Jayrot]

Originally Posted by missbunny

23 people sounds more right - I think I misremembered the 17.

You seemed to have misunderestimated.

[TellMeI'mNotCrazy]

Originally Posted by missbunny

I thought it was a much smaller number - in a group of 17 people at least two will have the same birthday more than 50% of the time. I think this was discussed here 2-3 years ago.

FYI, it means the same month and day - not the same year. The chances of that happening are much, much lower.

If it's a standard, elementary-school type classroom, the chances that it's the same year are pretty good, I'd think.

[Thudlow Boink]

Originally Posted by One And Only Wanderers

I find this hard to credit. I have never met anyone who has the same birthday at me, even when I was at University with a much larger class size than 30.

A common confusion. In a class of more than 30, there's a good chance that some pair of people have the same birthday, but you won't necessarily be one of that pair.

[Giles]

If you are only looking at the chances that someone else has the same birthday as you, you need to have a group of 253 other people for there to be more than a 50% chance that someone else shares your birthday.

(364/365)^252 = 0.500895161

(364/365)^253 = 0.499522846

That's assuming that there are only 365 days in a year -- to take into consideration leap year would be a more cmplex calculation.

[KP]

Actually, the calculation wouldn't be that much more complex. Instead of calculating using a base year, you'd calculate using a base period of four years +1 day (365*4 + 1 =1461) That's the denominator of your fraction. Since four of those days match your birthday, the numerator is 1461-4 = 1457. Since this is a small adjustment to your prior calculation, you can start using your prior result:

(1457/1461)^253 = .4997605448
(1457/1461)^252 = .5011325710

As you can see, the practical answer [integer # of days] doesn't happen to change if you consider leap years. The bracketing values remain the same. I did the calculation on a pocket calculator, which probably uses logarithms internatlly, and might be off on the lest significant figures. However, a little back-of the envelope calculation convinces me that it is certainly accurate to the first four decimal plaes, which is enough to verify the answer given.

For th OP, the first step is to calculate the chances that N people all have different birthdays. Ignoring leap years (which may not affect the answer anyway), we can do this by calculating p(x) - the chance that adding an xth person will not cause a match. Then we multiply all the p(x) from x=1 to x=N to get P(N). [For each value of x, multiplying the lower results gives the chances that you will reach the xth person without a match, and p(x) is the chance that the xth person won't *be* a match -- which gives you P(x) the chances that there is not match in x people (note the distinction between p(x) and P(x) - upper/lower case)

p(x) = ("allowable [nonmatching] days" remaining) / (days in the year)
= [365-(x-1)] / 365

P(x) = p(1) * p(2) * p(3) ... * p(x)
= ( [365-0] * [365-1] * [365-2]... 365-[x-1]x) / 365^x
= (365!)/[(365-x)!] / 365^x
= 365! / [(365-x)! * (365^x)]

You an do the same calculation, factoring in leap years, by changing the fraction p(x) to relect the number of nonmatching days in a 4-year period

Actually, the probability of birth is not evenly distributed throughout the year. Some month consistently have more births than others. This concentrates a disproportionate number of births in a "target rich environment" where collisions are more likely. I don't know, offhand if this effect would be enough to lower the integer solution to 22 in real life. It might depend on the country, since birth cycles through the year depend on cultural and climate conditions [among other things]

[Mississippienne]

Real Life Anecdote: In my piano class of 15 people, I discovered that another girl shared my exact same birthday -- December 6th, 1984. What are the odds of this? 15 is quite a smaller number than 30 or 50.

[TimeWinder]

Originally Posted by Mississippienne

Real Life Anecdote: In my piano class of 15 people, I discovered that another girl shared my exact same birthday -- December 6th, 1984. What are the odds of this?

100%, apparently.

Seriously, though, this is just confirmation bias, a case where you'd only notice a positive effect. Consider the tens of thousands of 15-people groups you've been in during your life, the vast majority of which did NOT have a person with you exact birthday in them...and you never noticed.

[Mississippienne]

Originally Posted by TimeWinder

100%, apparently.

Seriously, though, this is just confirmation bias, a case where you'd only notice a positive effect. Consider the tens of thousands of 15-people groups you've been in during your life, the vast majority of which did NOT have a person with you exact birthday in them...and you never noticed.

I don't care a whit about confirmation bias, I was asking what the odds were for 2 people in any random grouping of 15 to share the exact same birthdate.

[ultrafilter]

Originally Posted by TimeWinder

100%, apparently.

Seriously, though, this is just confirmation bias, a case where you'd only notice a positive effect. Consider the tens of thousands of 15-people groups you've been in during your life, the vast majority of which did NOT have a person with you exact birthday in them...and you never noticed.

This puts it the probability at about .25.

[Pushkin]

The first time I met someone who shared my birthday was at age 24, an Australian who was travelling Europe and had settled down to temp in the UK.

[jawdirk]

My Dad teaches math at a state college, and at the beginning of every class he bets the class a case of cola that two people in the class share the same birthday. Needless to say, they always take the bet, and he nearly always wins.

[Qadgop the Mercotan]

Originally Posted by missbunny

23 people sounds more right - I think I misremembered the 17.

It's the old 17/23 correlation. The Master spoke of it here.

[Finnie the Pooh]

22 people work at our maintenance department. Three of us have january 26 as our birthday. Three different years, though.

[clairobscur]

Originally Posted by ultrafilter

This puts it the probability at about .25.

When he wrote "100%" the poster meant that since it's already an established fact, it's probability is 100%.

[ultrafilter]

Originally Posted by clairobscur

When he wrote "100%" the poster meant that since it's already an established fact, it's probability is 100%.

I know that. But what does confirmation bias have to do with it?

[moggyAU]

hmm i dont think its that rare, i have a friend who has the same birthday as his brother and the were bron 2 years apart.

I have the same birthday as my aunt
My gf has the same birthday as my best mate

in fact i know many people who share the same birthday, some born in the same hospital on the same day and same year

[snailboy]

I've never met anyone with the same birthday as myself (although I hear Cher shares it). On the other hand, I've never went around asking my classmates their birthdays.

[t-keela]

Originally Posted by legion

I read somewhere once that in a class of 30 children the chances of 2 of the children sharing a birthday are about 50/50.

Is this correct and if it is, why is it?

First of all. A classroom of kids is not a random sample. Since it is a classroom the odds are nearly 100% that they will be the same age. So the birth year is practically a given. There are other restrictions in a classroom setting which increases the odds of an exact month. Cuttoff months and days for advancement as an example. I don't recall the exact month but there is usually a specific date that is the cutoff date for the age at which a child can start school.


All of that aside...
the biggest variable which can be applied to even a random sample is the fact that there are times when people simply have sex more often than others. Like Valentine's day for instance. Practically every new married couple is going to have sex on Valentines Day. Combine all these things and the odds are increased that two kids in a classroom will have the same date of birth.
somewhere in mid-late October. Then there's June newliweds and Christmas etc. but V-tines is the biggie.
In my own family (extended as well) there are at least a dozen birthdays that month.

[t-keela]

Sorry..that should've been November...you know about nine months after Valentines day sheesh...
sleep..get some sleep..

[t-keela]

I meant to add ...I remember why I was thinking October now. There's another big cluster nine months after New Years Eve. A really big cluster because of so many folks getting drunk and romantic etc...
and forgetting to do anything about their birth control.

[Bryan Ekers]

Originally Posted by Mississippienne

I don't care a whit about confirmation bias, I was asking what the odds were for 2 people in any random grouping of 15 to share the exact same birthdate.

The 25% figure doesn't measure an exact match, just from the 365 dates of a (non-leap) year. For an "exact" match, you'd have to compile demographic information and make the assumption that the people in the group are selected truly randomly (i.e. they not members of a class or peer group or anything else which would put them all of a particular age).

At this point, it gets exceedingly complicated, as multivariate statistics loves to do, so I'll just add some extra assumptions to make it workable:

• Everyone is American
• Everyone in the group is known to be at least 20 years old and less than 70 years old

Relevant informtion from the 2000 census, dividing the U.S. population into groups by age:

20- 24    19,185,063     
25- 29    19,316,817   
30- 34    20,587,073   
35- 39    22,648,354     
40- 44    22,535,368   
45- 49    20,230,558     
50- 54    17,790,616     
55- 59    13,559,151  
60- 64    10,864,730 
65- 69      9,533,955

These people have birthdays across a fifty-year span, representing 18,264 birthdays. Trouble is, the numbers show that these birthdays are not evenly distributed among this population. The 50+ groups show a marked decline in numbers (for obvious reasons). So just for laughs let's eliminate them and assume everyone in the sample is 20-49, whose numbers are reasonably consistant.

20- 24    19,185,063     
25- 29    19,316,817   
30- 34    20,587,073   
35- 39    22,648,354     
40- 44    22,535,368   
45- 49    20,230,558

This thirty-year span now comprises 10,963 birthdays, and we'll assume for simplcity that they're evenly distributed (they obviously aren't, but bear with me).

The odds of two out of 15 randomly-selected persons sharing an exact birthday are as follows:

1 - (10963!) / (10963-15)! (10963¹⁵)

~= 1 - 0.9905

= 0.0095, or less than one percent.

You'd have to sample 124 people before the odds of getting a match were more than 50%, which I'll admit sounds low, but the math seems accurate enough.

[Pushkin]

Originally Posted by t-keela

I meant to add ...I remember why I was thinking October now. There's another big cluster nine months after New Years Eve. A really big cluster because of so many folks getting drunk and romantic etc...
and forgetting to do anything about their birth control.

Yeah, the ozzie temp who shared my birthday said we were probably both conceived around New Years or possibly on it. Not that I like to think when or where my parents are at it, but nice to know that I was born of good spirits as the ozzie chick put it

[vanilla]

I don't believe it at all.

I have our church directory, which lists everyones birthdays.
There are about 160 members.
No one has the same birthday.

[Cabbage]

vanilla, the chances of that happening are 0.000000000000000099671433849300578506753494359707 %. To be honest, I don't believe you (unless your church has some strange policy, like not accepting new members if they have the same birthday as an already existing member). Maybe you should check the directory again.

[Cabbage]

But it just occurred to me, Vanilla, that maybe you're not understanding the problem. The claims made in this thread are not about someone having the same birthday as you; the claim is that, out of the 160 people in your church directory, there will be two people with the same birthday (month and day, not necessarily year)--you, however, might not be one of those two people.

[missbunny]

Originally Posted by Qadgop the Mercotan

It's the old 17/23 correlation. The Master spoke of it here.

My secret has been revealed!

You better watch out, QtM. "We" have our agents everywhere, even in that prison of yours.

[Qadgop the Mercotan]

Originally Posted by missbunny

My secret has been revealed!

You better watch out, QtM. "We" have our agents everywhere, even in that prison of yours.

Yeah, I met one of your agents earlier this week. I removed two of his toenails! He's now seen the error of his ways.

[Bryan Ekers]

Originally Posted by Cabbage

vanilla, the chances of that happening are 0.000000000000000099671433849300578506753494359707 %. To be honest, I don't believe you (unless your church has some strange policy, like not accepting new members if they have the same birthday as an already existing member). Maybe you should check the directory again.

I dunno, one-in-a-quadrillion events do happen, like forming a lasting healthy romantic relationship with someone you met at a dance club.

Just kidding. Barring a specific no-matching-birthday policy, and even accounting for some possible bias (i.e. babies in that particular town who are born during certain months have greater chances of dying in infancy, i.e. in harsh winter conditions), it's much more likely that there is a data error somewhere.

[Bryan Ekers]

Whoops! If babies during certain months are more likely to die, than the odds are even longer. The kind of bias that might spread out the birthdays of a population could be from there being only one local obstetrician who uses various medical methods to delay or induce labour in his patients to make it unlikely he'll ever have to deliver two babies in one day, plus he doesn't work Mondays, wednesdays of Saturdays, or some damn thing.

[Faldage]

Just a data point or three. I am in a group of 42 people with three pairs of same month/date birthdays.

[Fuji Kitakyusho]

My boss and I (the only two people in our company) discovered after working together for six months that we were both born on October 1.

What are the odds?

[Bryan Ekers]

Originally Posted by Fuji Kitakyusho

My boss and I (the only two people in our company) discovered after working together for six months that we were both born on October 1.

What are the odds?

One in 365, assuming the key element is that you have the same birthday, not that it's Oct. 1 in particular.

[Punoqllads]

Originally Posted by Fuji Kitakyusho

My boss and I (the only two people in our company) discovered after working together for six months that we were both born on October 1.

What are the odds?

Technically, 100%.

[TJdude825]

Originally Posted by Bryan Ekers

One in 365, assuming the key element is that you have the same birthday, not that it's Oct. 1 in particular.

What if the Oct. 1 part is important?

Before Fuji told us anything, the probability that he had a birthday of Oct. 1 was 1/365. The probability that his boss did was 1/365. The probability of both of these (1/365)^2 = 1/133,225 = 7.51 * 10^-6

In laymen's terms: Very small.

[Bryan Ekers]

Originally Posted by TJdude825

What if the Oct. 1 part is important?

Before Fuji told us anything, the probability that he had a birthday of Oct. 1 was 1/365. The probability that his boss did was 1/365. The probability of both of these (1/365)^2 = 1/133,225 = 7.51 * 10^-6

In laymen's terms: Very small.

Yes, but that's true of any two birthdays. Say Fuji's birthday was May 7th and the boss's was August 23rd. What are the odds of these two people having those birthdays?

1 in 133,225.

[MartinL]

Originally Posted by Fuji Kitakyusho

My boss and I (the only two people in our company) discovered after working together for six months that we were both born on October 1.

What are the odds?

I don't know, but: Happy Birthday!

[t-keela]

So let's see...Fuji, his boss, Pushkin and the temp, all conceived on New Years...
I wish I still had a distribution chart on birthdays. Sounds like there might've been some partying going on in the world that night..hmm.
but next morning...

I've got two nephews and a couple of cousins with birthdays this weekend.
My grandfather too, may he RIP.

oh yeah, happy birthday y'all

[TellMeI'mNotCrazy]

Well since the question has pretty much been answered I'll throw in some neat birthday coincidences that have occured in my life:

(all are different years)
My sister and I have the same birthday - hers 10 years after mine.
My father and ex-mother in law
My brother and ex's grandfather (with whom ex and I lived)
My daughter and my older sister

Those aren't big stretches - no fifth cousins fourteen times removed. Makes everyone's birthday easier to remember though.

[muttrox]

Originally Posted by t-keela

First of all. A classroom of kids is not a random sample. Since it is a classroom the odds are nearly 100% that they will be the same age. So the birth year is practically a given.

There are other restrictions in a classroom setting which increases the odds of an exact month. Cuttoff months and days for advancement as an example. I don't recall the exact month but there is usually a specific date that is the cutoff date for the age at which a child can start school.


All of that aside...
the biggest variable which can be applied to even a random sample is the fact that there are times when people simply have sex more often than others. Like Valentine's day for instance. Practically every new married couple is going to have sex on Valentines Day. Combine all these things and the odds are increased that two kids in a classroom will have the same date of birth.
somewhere in mid-late October. Then there's June newliweds and Christmas etc. but V-tines is the biggie.
In my own family (extended as well) there are at least a dozen birthdays that month.[/quote]

[muttrox]

Premature eposteation.

Originally Posted by t-keela

First of all. A classroom of kids is not a random sample. Since it is a classroom the odds are nearly 100% that they will be the same age. So the birth year is practically a given.

Birth year is irrelevant, this phenomona only concerns itself with month and year.

There are other restrictions in a classroom setting which increases the odds of an exact month. Cuttoff months and days for advancement as an example. I don't recall the exact month but there is usually a specific date that is the cutoff date for the age at which a child can start school.

Again, competely irrelevant as long as each year does in fact encompass all 365 days.

the biggest variable which can be applied to even a random sample is the fact that there are times when people simply have sex more often than others. Like Valentine's day for instance. Practically every new married couple is going to have sex on Valentines Day. Combine all these things and the odds are increased that two kids in a classroom will have the same date of birth.
somewhere in mid-late October. Then there's June newliweds and Christmas etc. but V-tines is the biggie.

Please provide a cite that this is a large scale phenomona. Considering the other myths about when people have sex (the great NY blackout ferinstance), this needs to be backed up.

[CurtC]

Originally Posted by Fuji Kitakyusho

My boss and I (the only two people in our company) discovered after working together for six months that we were both born on October 1.

What are the odds?

You have to be more specific about what exactly you want the odds for. If you're asking what are the odds that a given two people whare the same birthday, then it's 1 in 365.

The odds that you both share the same birthday, and that this date is October 1, is 1 in 133225.

The odds that you will eventually have a boss sometime in your life with the same birthday as you is, what, 1 in 15, depending on how many bosses you'll have.

The chance that you will, at some point in your life, be paired up with another person who shares the same birthday is very high.

When I got out of college and got a real job, I was surprised to find out that there were three others in the building (out of maybe 150 people) who shared my birthday (four of us total), and my secretary and I were born on the same exact day.

[Crandolph]

Originally Posted by moggyAU

in fact i know many people who share the same birthday, some born in the same hospital on the same day and same year

... and how are the quintuplets these days?

[guito13]

I share the same birthday with my sister 3 years apart...and my daughters have the same birthday.....but....they are twins :-)

[t-keela]

Originally Posted by muttrox

Please provide a cite that this is a large scale phenomona.

A cite that couples in the US get romantic on Valentine's day?
A cite that the world celebrates on New Year's Eve?

You're kidding right? These things need to be proven to you.