# Birthday stats

Even though there are only a few tens of people listed on my birthday calender, no less than 4 of them share my date of birth (i.e. day and month - not year). This seems an unlikely high number to me.

According to my (possibly dodgy) calculation, I would have to know the birth dates of about 365 x 4 = 1460 people to make it statistically likely to identify 4 people who unwrap gifts on the same day as I do.

How many people do you know of that share your birth date? Have you noticed the same trend, or is my case just a fluke?

Then I heard another interesting thing from a reliable source - well as far as unsolicited emails go - that if you take a class of 32 kids, the chances that at least any 2 of them share a birthdate is as high as 50% !

Mintaka

A girl I had a crush on in High School shared my birthday which was kinda cool

And today my wife and I share the same birthdate That makes it fairly easy.

Other than that I can’t recall any sharing my birthday…

It’s actually a 50% probability when there is a sample of 23 random people, at 30 the probability increases to 70%.

I do not know anyone personally who shares my birthday. Although, Valentino Rossi was born on the same day, but that’s nothing even related to this discussion because I don’t actually know him. (I thought that was true based on my brother’s encyclopaedic knowledge of bikes, but it turns-out that it is not even close).

But the main difference here between the probability of any random two people sharing a birthday (any birthday) and people who share your particular birthday is radically different.

The chances that I would share a birthday with someone in particular (say you, Mintaka, for example) is 1 in 365, the chance that we would both share it with a particular third person (say bluegrayV) is 1 in 133 225. A particular fourth person would be 1 in 48 627 125. This is only because I don’t know your birthdays. If I do know your birthdays, and they are the same then the probability is 1 in 1. Odd, huh?

ETA: There’s an experiment on this at Richard Wiseman’s blog, here.

James

Is that four people in addition to yourself, i.e. five people sharing the same birthday? If you give me the total number of birthdays on your calendar, I’ll work out the probability of four (or five) people sharing a birthday in that number.

As James has pointed out, in a group of 23 people there is a slightly better than even chance that two of them will share a birthday. This surprisingly low number is, unsurprisingly, known as the “birthday surprise” or “birthday paradox.”

'Luthon64

None in my case, however I’ve noticed a tendency of familial birthday’s being clustered in same months. e.g my dad, brother, my eldest son and I are in April, whereas my Mother, my youngest son and second brother are in June.

Is that four people in addition to yourself, i.e. five people sharing the same birthday? If you give me the total number of birthdays on your calendar, I'll work out the probability of four (or five) people sharing a birthday in that number.

5 people including self.
95 birthdays on the calendar.

Stop press: 'Luthon don’t bother. My dear wife just “admitted” to the crime of fudging the statistics. Every 3 years or so, she takes it upon herself to transcribe the old birthday almanac onto a new one . Whilst doing so she adds new people and delete those dead or out of contact, and the most obnoxious family members. Apparently, she never discards anyone who shares our own birthdays. Don’t ask me why. I usually don’t take much interest in birthdates or birthday almanacs, but this latest edition was hung - somewhat garishly - behind the door ( yes, THAT door) and I couldn’t help noticing what seemed like a huge unlikelyhood.

Mintaka

At my work there’s the usual thing where the birthday person brings cake in to the office. To me it’s strange since it’s their birthday, why should they buy cake for everyone? But I digress…

The month of July, we go into a special set of rules, since it seems a whole bunch of people here all have their birthdays in July (not me though, damn). And thus they get to “share” cake-buying trips to ensure everyone in the office doesn’t get morbidly obese.

I would (with my weak evidence) contend that things like the seasons, certain public holidays, big blackouts, etc. etc. Would all contribute to the distribution of birthdays not being uniform. Thus the chance of a random strangers’ birthday being the same as yours, would depend on what your actual birthday is to begin with.

The month of July, we go into a special set of rules

That’s me - dare I say - cancerian !

At my work there's the usual thing where the birthday person brings cake in to the office. To me it's strange since it's their birthday, why should they buy cake for everyone?

Same tradition in our labs. Although I notice a recent trend: people taking a day’s leave. I hope its not just to dodge bringing cake. Inexcusable!

Thus the chance of a random strangers' birthday being the same as yours, would depend on what your actual birthday is to begin with.
Might very well be the case.

Mintaka

Yes, this is true, as the article I provided a link to makes clear. Birthday distributions also vary geographically and according to culture. However, globally, many of these local biases cancel out, and the assumption of uniform birthday distribution is a reasonably good one. It also dramatically simplifies the complexity of calculating theoretical probabilities of birthday collisions, giving more than decent estimates…

Anyway, in a group of 96 people (I’m assuming the count of 95 excludes yourself for obvious reasons ), the calculated probabilities are given in the table below. Note that the general formula is pretty cumbersome, difficult to typeset properly here, and in any case beyond the capabilities of an ordinary calculator. You’ll need something like Matlab or Mathematica to do the actual evaluation. I also ran a Monte Carlo simulation with a sample space of over a hundred million birthdays in order to validate the formula.

[tr][td] No. of People Sharing a Birthday [/td][td] Probability (%) [/td][/tr]
[tr][td][center]1
2
3
4
5
6
7[/td][td]0.000 093
99.999 2
57.690 1
5.169 29
0.269 182
0.011 024
0.000 140 [/td][/tr][/center]

This puts the probability that five or more people in a randomly selected group of 96 individuals will share a birthday at about 0.280 345%, or around one in 357. That’s quite rare, but perhaps quite a bit more likely than intuition would at first suggest. Note also that not finding a shared birthday in such a group of 96 occurs less than once in a million, that having exactly two individuals born on the same day of the year is almost a dead certainty, and that having three or four same-day birthdays occurs in about two out of three such groups of 96.

Surprised?

'Luthon64

Only person that I know that I share my birthday with is my grandmother. And my sister and our cousin share a birthday.

I have noticed that a lot of people at work have birthdays in September, 9 months away from the December holidays.

I share my birthday with Isaac Newton - if you use the right calendar that is :

I know a lot of people with birthdays in October - probably for the same reason

BTW. You can now do this: http://forum.skeptic.za.org/tests/latex/

Ooops, only caught this tonight. Still, it made for an interesting and revealing exercise.

Thanks! We gots LaTeX! Now we confuzzles the bejabbers out of everyone. ;D

'Luthon64

Still, it made for an interesting and revealing exercise.

For sure - I certainly learned a few things about intuitive and real probabilities! Thanks for crunching the numbers for us.

ETA: Just one thing though, shouldn’t the probabilities in the table add up to 100?

Mintaka

No, because, for example, the probability of exactly two same-day birthdays does not exclude another pair or triple or quadruple, etc., of birthdays occurring on a different day in the same group of 96. All that is asked for is what the probability is that at least one pair or triple or quadruple, etc., of same-day birthdays occurs in a group, irrespective of any other same-day birthday occurrences in that group. Thus, a given group may fall into more than one of the categories given in the table, and hence the probabilities add up to more than 100%. This effect increases as the group size increases because the number of possible overlaps increases exponentially with group size.

(It should be noted that the last value of “7” in the first column of the table should actually be read as “7 or more” — something I neglected to point out at the time of posting.)

'Luthon64

Now tell me what combination of numbers stand the best chance for Saturday’s lotto

All of them.

That would be any number that wasn’t a winner last week

The following table lists the probabilities associated with having at most n people sharing a birthday in a randomly chosen group of 96:

[tr][td] Max No. of People Sharing a Birthday [/td][td] Probability (%) [/td][/tr]
[tr][td][center]1
2
3
4
5
6
7+[/td][td] 0.000 096
39.771 5
54.774 1
5.175 18
0.267 494
0.011 225
0.000 399 [/td][/tr][/center]

In this case, the probabilities add up to 100% because any given group of 96 can only fall into one of the categories, but note that it does not exclude, say, in the row labelled “4” three distinct occurrences of four people sharing a birthday.

As for Saturday’s Lotto numbers, I’ve been sworn to secrecy so I’m not telling, and this thread should immediately be moved to the Conspiracy Theories board. Nevertheless, you would do well to try a combination of seven numbers, each between 1 and 49, so that no number is repeated… :

My honest advice? Play Quick Picks only because they never repeat number combinations for a particular draw, and that’s countrywide. If you do then win the jackpot, it’s unlikely that someone else will have chosen the same combination of numbers so you won’t have to share the prize.

To think that any previous Lotto draw’s results can affect the present one is to commit the so-called “Gambler’s Fallacy”. It’s amazing how many people actually do just that. In fact, just about everyone with a Lotto “system” is guilty.

'Luthon64