Sure, same order of magnitude, I just felt like the number should be significantly less. Like I would think a natural intuition would be that you’d need half of 365 for there to be a 50-50 shot of someone sharing your birthday, but it’s not 183, but rather 253, significantly more.
Thinking about it, I can understand why that is not the case — it just was a little surprising to me.
Not half, you’d expect ln(2)
All of these are just theoretical calculations; they are off from actuality in several ways.
First, they are using 1/365.25 in the calculations, as if every single day of the year had the exact same chance of being a birthday. Which is far from actual reality.
In North America, Christmas Day is a very infrequent day for births. By far, most births are in late summer - fall (July-October), reflecting conception happening in mid-winter thru early spring. Even weekly, it varies: births are more common mid-week (Tuesday-Thursday) and rarer on weekends, especially Sundays. [This may be due to doctors/hospitals schedules, since 20% of births are induced and 33% are C-sections.] And it’s a very significant variation; there are literally twice as many babies born around Labor Day as on Christmas Day.
Also, these calculations assume that the ‘people in the room’ are an independent cross-section of the population. Obviously, they aren’t – they are all gathered together in one room, presumably for some common purpose. Like the educational classroom examples: all the people will be about the same age. Similarly if you pick 12 people at a private dinner, or 12,000 people at a public concert – all likely to be of a similar age. Also residing in a similar area, of similar educational level, economic background, probably even similar political views. And for a small group like a dinner party, probably some of them are related, either biological relatives or related by marriage(s). So far from an independent sample.
So while this ‘birthday problem’ is a fun item, and a good illustration of how for most people statistics refute their ‘common sense’, these statistical calculations don’t strictly match reality.
The “2 people sharing a birthday” odds was our first lesson in my finite mathematics class in college. As it happened, despite having enough people to put the odds over 0.5, there were no shared birthdays, so it was a little awkward for the professor, but he still showed us the math to underscore that “intuitive” answers are often wrong.
Yes, but these calculations are fine for a first order approximation. I would be surprised if the actual “birthday” problem odds differ significantly from the theoretical answer of 23. And, if it does, one should think that the answer is actually 21 or 22 (as some birthdays are slightly more likely to show up), so even more common and surprising than the 23 answer.
Yes, if you have a strong mathemetical/statistical background. If you are a lay person, I think you’d intuit around 183, especially as most people have no idea what a natural log is. That’s precisely also why the answer to the birthday problem is so surprising to most. If you know your math and look at it, you can see why it makes sense. But I doubt the vast vast majority of the populace would guess anything around 23 if they have never heard this problem before.
It really shouldn’t be any more awkward than throwing a heads when calling tails. I’m sure the professor expects this not to work about half the time (assuming a class size around 23.) Meanwhile, in a class of 41, we’re up to a 90% chance.
Different but related birthday probability question:
Given two people chosen at random, what is the probability that they have the same birthday? (At first thought, it seems like approx. 1/365. Is that right?)
The situation I’m thinking of is this: Given a married couple, who probably chose each other without regard to their birthdays, what is the chance of them having the same birthday?
Yes, so around 0.27%