## Sunday, 20 December 2009

### Coincidence and Probability

When I posted about Emila's 2010 Illustrated Calendar a few weeks ago, blogger Wan Lili of Suddenly, Heta! commented that she shares the same birthday as me. I was pleasantly surprised and noted that the chances of that to happen is quite low because this blog of mine doesn't really have that big a following.

Lili's comment reminded me of a programme I saw on The Discovery Channel some months back, about the probability of finding two persons within a group of people who share the same birthday. This problem is also known as the Birthday Paradox and there are quite a number of articles about it available on the internet. The different articles describe the problem in different ways but perhaps the simplest way that I can put it is as follows :

What is the minimum number of people needed within a group so that the odds of finding any two who have the same birthday becomes 50%?

Even for those of us who are mathematically inclined, the initial assumption we arrive at is that it must be a large number. There are 365 days in a year (ignoring leap years for simplicity)... and for 100% probability (i.e. a sure thing), we need 365+1=366 persons. Therefore, for an even chance of finding two people with the same birthday, the number is 50% of 366 or 183 persons.

This answer is wrong. Probability theory shows that we only need to gather 23 persons for the odds to become even (i.e. 50:50 chance). Now how can this be? I don't want to bore you with the mathematical analysis of this problem but you can read the links I've included below for detailed explanations.

This phenomenon is not actually a paradox in the logical sense of the word but it seems so to most people because the mathematical truth contradicts natural intuition. The human brain thinks of progression and extrapolation generally in linear terms and gets confused when some things expand on an exponential basis.

In the second reference article below, Robert Matthews and Fiona Stones carried out a study to test this theory by looking at football matches. In a football match, you can find 23 people i.e. 11 players from each team and the referee. If there are 10 such matches, then probability theory says that we should be able to find birthday pairs in 5 of them (50:50 chance). Matthews and Stones did their analysis on 10 Premier League fixtures played in 19 April 1997. It involved them checking the birthdays of 220 players and 10 referees. Sure enough, they found that there are coincident birthdays in 6 of the matches. In fact, they found two pairs in 2 of the fixtures.

To extend slightly on this subject, the theory also says that we need only 57 people for the probability of any two people with coincident birthdays to become 99%. In other words, if there is a gathering of around 60 persons, I'm willing to bet that I can find at least one pair that share the same birthday.

Graph from Wikipedia, showing the approximate probability of at least two people sharing the same birthday amongst a certain number of people

I did a bit of my own analysis on this matter using the birthdays of my Facebook friends. I compared their birthdays against the sequence of when we became friends. The result is pretty close to the theory. The 24th person who became my FB friend shares the same birthday with the 9th person. I did not have to wait long for the second pair. The 28th FB friend has the same birthday as the 22nd friend.

I was about to do the same analysis on the extended family on my wife's side (parents, siblings, in-laws, nephews, nieces etc.) but then realised that the same-birthday occurrence is found even earlier. My father-in-law (the no. 1 guy) has the same birthday as his youngest son (the 15th family member).

To use the conclusion of the Matthews & Stones research, coincidences really are “out there”, as probability theory predicts, if we take the trouble to look.

References :

1. Birthday Problem - Wikipedia
2. Coincidences : the truth is out there

3yearshousewife said...

Complicated to me, the not so bright in mathematics. In my place of 5 M'sian families (32 people) 3 of them share the same birthday.

HLiza said...

Interesting..you really went out to do an research on this it seems..I've never found anybody who shraes my bday so far.

lili said...

Salam Oldstock,

Reading this entry very early this morning and seeing my name flashed in bold..hehe...I just can't help blushing silly!

In my family, I share my birthday with a nephew and you can imagine my excitement the day he was born. To this day, he is still attached to me even though he's married!. My elder brother shares his birthday with my niece; and a SIL with my another niece.

Okay, let's not delve any further, hihi...I will have to check the article you linked here.

-Wan Zalili

Zendra said...

Oldstock, fascinating info! Amongst my band of 30 nephews and nieces, I share my birthday with 1 other - 70% chance. Conclusion - there are 2 cool-cats in the family.

So what's the probability you and I share the same birthday?

mamasita said...

Snakebite said...

tak larat i nak baca sampai abis, berpinor biji mata. sebab tu lah i jadi lawyer, not an engineer ke accountant ke apa ke yang involves mathematical culculations, statistics, charts bla bla...cannot compute

Tommy Yewfigure said...

Aiseh oldstock, u memang think like an engineer or a kaki kopek :)This sort of thing really give u a buzz. Hey what about this? Good odds or not? Australia winning FIFA World Cup 2010 at 64:1 ?

FYI; now this is zen4U; I share the same birthday as one of Zendra's kid too...no joke!

Tommy

Oldstock said...

As,

Not many people like mathematics, that's why I left out the explanations.

3 people out of 32 sounds like good odds to me.

Oldstock said...

Hliza,

I like working with numbers although I'm not that good in remembering them. I did the analysis just to kill some time at home... masa tu tengah dok rehat kat rumah recovering from fever. Nak buat kerje manual lain takleh, jadi dok depan laptop, opened up a spreadsheet dan mula la buat keje yang tak berapa faedah ni, heheheh...

Oldstock said...

Lili,

As I mentioned previously, you and I will now remember each other for at least one day each year, for the rest of our lives. Now that's a fact that's really worth blushing for :-)

As you can see, you already have three pairs of same-birthdays in your family.

Oldstock said...

Zen,

As you noted, the chances increase when you add more people to the group. But in your case, if you can rank the family member that shares your birthday (in terms of age seniority), maybe the group could be smaller.

The chances of me and you sharing the same birthday? You kena bagitau birthday you dulu... :-)

Oldstock said...

mamasita,

It's July 6, 1962. Still young kan? Ehem...

Oldstock said...

Snake,

Tu belum lagi I bagi formula, P(A')=1-P(A) ataupun P(x)=1-P(Qn).

Numbers cannot lie... but those who manipulate them do so, heheheh. But given a set of facts, ten different lawyers can have ten different interpretations, muhahaha!

Oldstock said...

Aiseh Tommy... what to do, memang engineer like to have fun with numbers. But not good enough to be a kaki kopek, because you need a cool head for that.

64:1 for Australia to win the World Cup are not good odds (betting-wise I mean). I think the bookies are afraid of giving better odds. Should be more like 100:1 for me.

If I'm a betting man, I'd look at odds offered for who's NOT going to win. E.g. what's the odds for Spain not winning the World Cup, or what's the odds for North Korea not making it to the second round. But alas... my skills in this area is not polished yet ;-)

Tommy Yewfigure said...

Wah u r good; they've revised it according to ur guess at 101:1 for Oz, 5.50:1 for Spain; Brazil 6:1, England 6.50:1. NZ & N.Korea at 1001:1....maybe can put \$1 on them hahaha.

Aizan Suhaira said...

Too much maths....

Uncle Lee said...

Hi Old Stock, very interesting. Maths not one of my good sides, ha ha....but it sure is nice to know.
For me, not too keen on birthdays.
Sebelum 15 years okay....today the candles cost more than the cake, ha ha.
Wishing you the best of Seasons greetings, Lee.

Patricia said...

I'm not really one for numbers, but this post was fascinating, just the same. In my family of six, my mum and one sister share the same birthday. Cool, eh? Though it can't have been fun for my mum, all those years ago, to spend her birthday in a hospital!

You and I share a birth month, which is easier, as there are only 12 months. And I was six when you were born!

pakmat said...

..fascinating..and what are the odds of finding two persons who doesn't have a birthday..? like me..? there's no documentation to my birth..:)

Ok...in my book, you are now officially a geek!

And that's a compliment =)

masjunaida said...

Hi,

Interesting article this one !

So does the theory also say what is the minimum number of people needed within a group so that the odds of finding any two who have the same birthday becomes 100%?

Oldstock said...

Tommy,

Put \$5 on Oz and \$1 on Spain... good luck :-)

Oldstock said...

Aizan, alaa... baru sikit je tu.

Oldstock said...

Lee,

Don't worry about the candles on your birthday cake... the heat would keep you warm in the cold Canadian nights, heheheh. Season's greetings to you too.

Oldstock said...

Pat,

We are July babes... always cool.

Oldstock said...

Pakmat,

You are like my father, no birth cert. So the whole year is your birthday.

Oldstock said...

Tq. Memang dah lama jadi geek pun... cuma sekali sekala kalau nampak macho pun best gak kan :-)

Oldstock said...

Masjunaida,

Welcome to the blog... that contains useful and useless info all the same, heheheh.

Theoretically, to get 100% odds, then you need 365+1=366 persons. Meaning to say that in a random gathering of 366 people, we can definitely find at least two persons sharing the same birthday, because if by some stroke of luck, the first 365 persons all have different birthdays, then the 366th person must surely have the same birthday as one of the others.

But we need not go that far to reach very favourable odds. If you take a look at the graph I copied from Wiki, by the time we have 60 people, the odds are already 99.9%.

Hope I have not bored you with this explanation.

anneaziz said...

Hello! Calling for 19th July, anyone?

Seriously Oldstock, I've yet to meet many born on the same day as me. So far I've met only one and that is my sister's neighbor. Why is that? Is it that I don't mingle enough?

We should start a count for those born in July...seems like blogging is a favorite pastime!

Andrea Wh@tever said...

Mr Oldstock, you may be interested to know that this is one neat trick which Maths/Stats students love to play on the innocents and impressionables to get free drinks at the bar. :)

Oldstock said...

anne,

The day will come soon enough when you'll meet somebody who was also born on 19th July. As the Michael Jackson song goes... You are not alone.

Kira 6th and 19th dah tak jauh la tu...

Oldstock said...

Andrea,

Aah.. now why didn't I think of that, all those years ago...