I was doing some web searching and came across this article from 1994 by Mahmoud Sayrafiezadeh, which begins:
The birthday problem asks for the probability that at least two people in a group of k people will have the same birthday. . . . The present work was motivated by the need to provide an approximation formula for the solution of the birthday problem in a liberal arts course on the Nature of Mathematics. The main result enables students who have not yet studied calculus to approximate solutions to birthday-type problems.
So far, so good. But here’s the funny part. The article continues with some formulas:
OK so far, maybe. I suspect that this level of technical detail will already confuse the liberal arts students who have not yet studied calculus, but, who knows, maybe this person was able to teach it in the classroom.
But now things start to get really hairy:
At this point, I can’t believe that those liberal arts students who have not yet studied calculus are following. I don’t mean this as an insult to the students! Mathematics is a foreign language to most people, and it’s not a disparagement to say that non-math-speakers will be challenged by the above expressions, any more than it’s disparaging me to say that if you talk to me fast in Spanish, I’ll quickly get lost.
The article continues:
And a bit later it really gets going:
If this is the elementary version for the non-calculus-based liberal-arts course, I hate to think what this teacher did for the advanced classes. Maybe they’d prove the Riemann hypothesis as a homework problem?
Again, I assume there’s nothing wrong with the content of the paper; it’s just funny to think of someone teaching math to liberal-arts students and not being able to resist all this technical stuff. I guess most of us are like this when we teach!
P.S. I showed the above to someone who was a teaching assistant for math classes, and he said it reminded him of the famous xkcd quartz cartoon.
I have used the birthday problem for non-mathematically trained students by using a Monte Carlo simulation. It has the advantage of being able to use the actual non-uniform data on birthdays over the year (as well as not requiring the calculus in the computations above – which, in any case, cannot easily handle the actual birthday distribution). For liberal arts students it is far more intuitive. But if the point is to illustrate the “Nature of Mathematics” I guess it can be debated as to which approach is more appropriate (is it the nature of the subject matter or the nature of those who practice the subject matter?).
In the article he is developing the approximation that he actually gives the students. I don’t think he goes through the derivation in class.
That might be worse. Giving a fairly complex formula without the derivation is a strange way of demonstrating the “Nature of Mathematics.” In my mind, it is more likely to achieve shutting liberal arts students from further study or use of mathematics than providing any insight or motivation.
Also, the approximation formula is itself way too complicated for liberal arts students who have not studied calculus. It’s the sort of formula that’s hard to understand!
Also, the author writes of the “need to provide an approximation formula for the solution of the birthday problem.” But there is no such need! The brute-force formula is clear, demonstrates the probability reasoning, and is easy to calculate on any computer. What’s the point of the approximation formula anyway?
It says the calculators (in 1994) couldnt handle the straightforward combinatorics. Most code and math looks way more confusing than required to yield correct results due to need for computational efficiency. People end up stringing together a bunch of arcane tricks to get the job done in practice.
All of calculus, in fact, falls into this category. We don’t need calculus at all, but it makes approximations far more tractable.
Isn’t the paper written for the instructors? The instructor doesn’t have to use the same notation to explain the formula on the board. They can do it in a simpler way.
Can’t reply to Anoneuoid, but I’m pretty sure the calculator I had in the 80s had a button for factorials. 365! is probably too big for a normal calculator to handle but I think a TI-30 would give you an answer in scientific notation. The principle surely could be illustrated with smaller number and the instructor could supply the values for the birthday problem. Or the students could be introduced to the idea of looking values up in a table. If this was a class for students who might not have very much math at all, asking them to memorize and use any formula seems like a lot. Surely the point is to explain how combinatorics and probability work, not offer arbitrary formulas for a couple of specific problems. But as a computer science person, not a math person, of course I’d think that.
TI-30 and TI-36 had/have a limit of 69!. 70! gives an error.
As I wrote elsewhere in the thread, I think the best way to do the calculations is to multiply together the ratios, not to touch factorials at all. Brute force always, is my motto.
He says that the work is motivated by the need to provide an approximate solution. You could,
1. Show the students the exact solution and the limits of their calculators to compute it.
2. Explain that using mathematical tools one can obtain an approximation to such solutions that is easier to compute at the cost of some error. One could even show some equations here for illustrative purposes only.
3. Explain to the 1994 crowd that many problems have difficult to compute exact solutions and that one can use mathematics to obtain good approximations. Wouldn’t that be a good superpower to have?
I feel like Andrews reading of the paper is the least charitable reading. Paints the author as clueless when we don’t really know what he was doing in the classroom.
Not to disagree with the main point, which seems to be that subject matter experts overestimate what students/non experts know when explaining things. I just think that the authors exposition on an paper is probably not a good reflection of how he used the material in the classroom (I would assume).
Lee is correct, and to respond to Andrew’s comments:
(1) the approximation formula, Eq. 2, is not “way too complicated” for a liberal arts student. It’s just one fraction and an exponent, and it’s short. (Yes, that’s too complicated in 2025, but 1994 was a happier time…) Unless my quick skim is missing something, the rest of the paper is just the explanation of Equation 2.
(2) Again, in 1994, the brute force solution is inaccessible to most students. Computer access was rare! I was in college then, by the way, and sending students in a non-STEM class to the basement computing lab to write a program (?) to calculate a series sum would have been absurd.
(3) This is for Dale’s comment: one points interested students to the derivation of something like this; one does not force everyone to slog through it. Given e.g. 40 hrs of class time in a term, spending 1 on a long mathematical derivation would be a bad decision.
(4) The point of articles like this is to highlight some potentially useful teaching method or tool. Presumably, the author has actually implemented it, and found it useful. My only criticism of the paper is that it doesn’t actually describe this implementation.
It is hard for me to remember 1994, but the software I currently use for simulation (@Risk) first came out in 1987 and spreadsheet based simulation software was available in 1982 for Apple computers. My reason for citing this is not to disagree with what you have said – these are all good and valid points (I was unable to place myself in my 1994 world). But I do think it illustrates one unfortunate feature of academia – it lags behind technological developments. Too many things we teach today are reflective of a technological environment that existed when we were students and don’t match the realities of our current students. It does make me wonder about our approach to AI (and, please, let’s not digress into a debate about whether AI is a parlor trick or something more substantial – I am just making a point that academia adjusts rather slowly, e.g. witness the paucity of good intro stats books compared with the number of such books in their umpteenth edition of NHST).
Of course it is true that most students did not have computers in 1994, so your points about the relevance of the approximation formula remain valid. At the same time, by 1994 personal computers should have (arguably) made more inroads into academia than they perhaps did (I got my first in 1984 and started using Lotus 123 shortly afterwards, despite the fact that I had no interest in accounting and was only teaching economics).
Raghu:
First, the exact expression is not so hard to compute. No need to break out the factorials. Just compute (364/365)*(363/365)*(362/365)*… It’s 22 multiplications and 22 divisions; not so hard as all that, and then you can see the number gradually getting smaller. It also motivates the approximate solution where you just take the middle term in the product and take it to the 22nd power.
Second, I can believe that the author used this formula in class and that he was satisfied by it; I’m skeptical that the students got much out of it! But, yeah, you’d want to see what actually took place in the classroom. In my own articles on teaching (for example, “Two truths and a lie” as a class-participation activity) and in our Active Statistics book, I go into lots of those details of implementation.
> I’m skeptical that the students got much out of it!
They got to calculate the result with a few operations. 22 multiplications and 22 divisions may not be hard but its one order of magnitude more operations than what the approximation requires.
Carlos:
Sure, but they don’t need to calculate the result! The point of the birthday problem is not the calculation of the result, it’s the surprising idea, due to combinatorics, that the probability of a shared birthday is so high. There are various ways to convey this (my favorite is the approach noted by Dmitri elsewhere in this thread and described in Mosteller’s book from 1965). I don’t think the formula presented in the paper discussed in this post is helpful for this purpose, as it will just be mysterious for those non-calculus liberal arts students.
> Sure, but they don’t need to calculate the result! The point [is] that the probability of a shared birthday is so high.
If you tell a class of 30 students that the probability that there is a shared birthday is 1 − (365/365)(364/365)(363/365)(362/365)(361/365)(360/365)(359/365)(358/365)(357/365)(356/365)(355/365)(354/365)(353/365)(352/365)(351/365)(350/365)(349/365)(348/365)(347/365)(346/365)(345/365)(344/365)(343/365)(342/365)(341/365)(340/365)(339/365)(338/365)(337/365)(336/365) the fact that it’s so high could be lost on them.
If you add that the result of that calculation is approximately 70.5% it might help them to get the point, it surely would help me.
Carlos:
If I were doing this example in class, I’d demonstrate the computing in R. If I’d been doing it in class in the pre-computer era, I’d write it on the blackboard as (364/365)*(363/365)*(362/365)… and compute the probabilities ahead of time.
Computing the probabilities ahead of time is one option. An approximation can still be useful to compute the probabilities once the attendance count is known and it can be shared with students so they can check the probabilities for different group sizes as they wish. Of course you could also precompute a lookup table – and you could also share it with students – but it would be specific to the 365 case and students may be curious about other cases.
I wonder if Mahmoud Sayrafiezadeh’s interest in putting the means of calculation in the hands of the masses was part of his Trotskyist activism. He seems to be the father of Said Sayrafiezadeh, author of “When skateboards will be free”. I didn’t know about the book until I looked up the name, but it seems interesting: https://granta.com/when-skateboards-will-be-free/
Prior to college I took Calc, trig, etc…. and a special course called college math which should really have been advanced calculus. And I got it all.
Now, I went to The CItadel and was in the humanities. But we all have to take 2 semesters of math. Fine. My fault for not taking that AP exam.
Now the fun begins. Each semester was a class of… well its all non math students. You’d think…teach to your audience, improve their skills! And we’re all freshmen mostly so we spend most of the year a little shell shocked.
First semester I had a teacher who…to put it mildly, made me hate math. Professor R….would cover a significant problem each class and teach how to solve it. Fine so far… he’s got us twice a week for 75 minutes each. 5 min to set up, and off we go. So he’s walking through the concept as illustrated by a particular problem on the blackboard, explaining each step. Furious note taking occurs all arounds and everyone copies the work….
30 min in, he finishes, and proclaims… as you an see this isn’t the right answer, that method is wrong…and erases it all.
Okay, fine, we’ve got time. 5 min to set up again and away we go. 30 min of furious note taking and trying to understand, a little Q&A about what is being done… and an answer.
Wrong again. Erases it all.
Now he says, I’m going to show you all the RIGHT way to do this. We have 10 min.
He gets halfway thru, and class is over, and we all have to scurry to our next classes. Still not knowing how to do what he was supposed to teach.
The next time in class…its an all new problem. And we repeat the same failed process.
I scored in the top 5 % on the math section of the SAT in 89-90. And he undid everything I had learned up to that point. I dropped him for the next semester. That professor (major something, full professors got military rank) did teach the right way. in the allotted time. But you’d better be able to read what was on the board because he speech level was barely above a whisper and he mumbled.
I dunno, Equation 2 doesn’t look so bad to me. Divide, subtract, exponentiate… I’m not sure it’s actually useful for students at that level to have that in their toolkit (or maybe at any level! I’m certainly not going to bother memorizing it), but if the teacher wants them to try some things — “how many people would have to be in the group in order to have a 99% chance that at least two of them share a birthday”, stuff like that — then OK, maybe that’s fun or maybe the students really can gain some intuition into the situation.
I had a fellow-student in my 1st year graduate Psych stats course who was having a bit of a struggle. It seems that the last formal mathematics class he had had was in Grade 10 in Québec which would have made him ~ 16 years old.
I suspect lot of US liberal arts undergrads would have about the same level of math knowledge as my fellow-student did.
Thinking back to the distant past when I was in Gr. 10 in Ontario (again ~ 16 yo), an equation like that would probably induced panic in me. I imagine I could have decoded/deconstructed it with a bit of time and effort. Today it’s pretty straightforward but that’s after a couple of calculus and linear algebra courses in high school, a calculus course and a linear algebra course at university plus various stats courses at university.
Interesting. Every time I learned about the pigeonhole principle, it was taught using the birthday problem. I find it remarkable, given the efficiency of Internet searches that the pigeonhole principle and a simple iteration of the problem wasn’t either discussed or found. That is, this shouldn’t require calculus to do a trial and error search for the answer.
In any case — because no one has mentioned it — here is the basic derivation:
The probability of no match (that is, if there’s one person in the room) is: 365*365*…*(365-k+1)/365^k, where k is the number of people in the room. (Also, the 365-k+1 can be derived using the pigeonhole principle when k is greater than 365.) So, when k=1, there is no match. Then, let k=2, is there a match? No. Repeat until a match is found and the answer is k=23.
So, what should happen for students is that they’ll see that the percentage who share a birthday linearly increases with each value of k. So, by the time they’ve found a value of, say, k=5 or k=6, they’ll understand that the answer is much less than what intuition suggests.
Sorry, the formula should be: 365*364*363…*(365-k+1)/365^k.
There are n^2 ordered pairs in a set of n objects. For unordered pairs of distinct objects you have to subtract n to remove the duplicates like xx, and you have to divide the result by 2 to account for the two orderings, xy and yx, so (n^2 – n)/2. You want this number to be comparable to 365/2 so n ~ 20.
Dmitri:
Yes, this is the classic argument. Frederick Mosteller presents it well in his classic book, Fifty Challenging Problems in Probability with Solutions.
Oh, sorry about that! Speaking as a liberal arts guy it is way better than Sayrafiezadeh’s formula.
Dmitri:
No need to apologize. The n*(n-1)/2 argument is pretty obvious, no way it’s original to Mosteller, nor did he claim any originality in his book. His explanations and discussions are good, though.
Mosteller gives a good discussion, but it doesn’t seem to be about this ‘solution’ (perhaps there is another edition I don’t have at hand?). I write ‘solution’ as I don’t follow and can’t reverse engineer a real solution that looks roughly like this.
Jon:
Hmm, maybe I’m misremembering. I’ll take a look at the book next time I’m in my office.
If the idea is to explain *why* the answer is so far from intuition, the teacher could have used a small n. What’s the probability that two people were both born on Tuesday?
Cartoon:
It’s not whether they were both born on Tuesday, it’s whether they were born on the same day of the week. But I think the intuition-busting doesn’t work so well with n=7. The intuition comes because it’s natural to compare n to k, but you really should be comparing n to k*(k-1)/2. So to get a counterintuitive result, it helps for (k-1)/2 to be big. k roughly scales with sqrt(n), hence you need a big n (such as 365) to get a reasonably high k.
So, yeah, the math is kind of interesting, and I think it can be explained to liberal arts students who have not taken calculus. I just don’t think the formula and derivation in that linked paper is the way to go.
I tried to correct it but the site doesn’t allow comments on not-yet-approved posts.
I usually gravitate toward small n when trying to make things intuitive but maybe that’s not the right approach here. Another idea: point out that in a group of 367 people p = 1, so surely if n is *almost* 367 p can’t be much less than 1, can it?
Hmm. I’m not sure I understand your critique of the day of the week analysis.
It seems to me that the paradox is that, thinking quickly, one might think that somewhere in the neighborhood of N/2 people have to be in the room for a collision to have 50% chance of occurring. But the correct answer is more like sqrt(N). The problem with n=7 is that 7/2 is close to sqrt(7) (3.5 vs 2.65). Doing the case N=7, I get that the probability of a collision exceeds 0.5 if there are k=4 people in the room. But, 4 is the first integer greater than 7/2, so the N/2 intuition is confirmed in this case!
I don’t think you want a reasonably high k, rather I think you want N/2-k to be large—thus creating the “paradox”.
Perhaps what you mean when you say a reasonably high k is that as k gets bigger, the disparity between k and N/2 grows.
Am I missing something?
The birthday problem is a way to introduce important ideas in a “Math for Poets” class. Proofs no matter how carefully and clearly done, will not convince those students. They don’t think that way so don’t waste their time. They are often not convinced even with empirical demonstrations. Important ideas like the probability of a complementary event, the probability of two independent events, recursion, sequences and multiplication of fractions between 0 and 1 can all be used in the discussion of the birthday phenomenon. Once introduced, I liked talking about the probablity of recurring independent events with high probability of success – the classical example from reliability theory. Again a “surprising” result that can be explained with the birthday problem concepts. b-t-w How many people use the Central Limit Theorem with no idea of its proof or for that matter not enough math background to understand its proof.