1. ## Birthday Paradox Math Project

I am currently taking precalc and have a math project due. After flipping from topic to topic I finally settled for the birthday paradox. I see that the solution for the birthday paradox itself may be too simple for a project, but I found this paper that also discusses the approximation of the birthday paradox. Probability isn't one of my strong points, and some of the explanations are confusing for me:

1) There is an intuitive reason why the probability of matching birthdays is so high. The probability that a given pair of students have the same birthday is only 1/365. This is very small. But with around two dozen students, we have around 365 pairs of students, and the probability one of these 365 attempts will result in an event with probability 1/365 gets to be about 50-50. With 100 students there are about 5000 pairs, and it is nearly certain that an event with probability 1/365 will occur at least once in 5000 tries.

*I don't understand this explanation...how is it that 24 students is equivalent to 365 pairs of students? The statements afterwards confuse me as well

2) I am interested in adding the approximation part into my project but I have no idea what an approximation really is for... what is the difference between it and the solution? This is the link to the paper..birthday problem starts on page 42:

http://ocw.mit.edu/NR/rdonlyres/Elec...AC2/0/ln10.pdf

-Being in high school, am I over my head with the approximation part? If not, do you have recommendations of what I should research to help myself better understand it? for ex. I don't understand what 1-x = e2 is for.

I know this is a lot of information and a mouthful to read but any help would be greatly appreciated

2. That's not what they are saying. Imagine that you have 24 students. How many ways can you pair such students? 276 ways. Not quite 365 but kinda sorta close. So they are saying that even with 24 students, you have 276 chances that there is going to be a pairing that has the same birthday. And the entire EVENT is that if you go through the entire class you'll find a pair of birthdays, not if you take two people from each class and determine if they have the same birthday. Does that make sense?

For the approximation, you really aren't meant to "understand" anything except how to type in and calculate that formula. They throw a few terms out at you that probably aren't going to mean anything to someone just in pre-calculus. All you need to know is how to calculate the size of your group to get the rough approximation that you want. At the end of their section they used .5 and found that with 365 days, you would need 22 students to get a roughly .5 chance that at least two students (one pair) have the same birthday.

How are you trying to formulate your report? Out of that entire section ther are about two parts that are probably relevant - you'd probably have to explain why some of this stuff works, which would have you going into topics you may or may not have covered in HS.

3. That's why I felt that the explanation was confusing...that paragraph I posted was an excerpt from that paper. With the combinations formula I got 276 as well, but the paper said 365.

So the approximation is used by plugging in a percentage and finding the number of people you need to get that percentage?

Well supposedly this is supposed to be in the format of an IB math project since my math teacher is the IB math teacher (though I am not taking Ib math right now...she still gave us this project) In other words, it's supposed to have an introduction explaining what my investigation is, a step-by-step process explaining how I intend to do it, the actual mathematics part, conclusion, validity.

I was planning to show how to get the solution and then maybe add in the approximation because I felt that without it, the project would be too simple. I also am planning to test it with data by getting some birthdays like from sports teams online to test the birthday paradox. (because I should have some data associated with my project based off of my rubric)

I really want to understand the birthday paradox (especially how they derived that approximation formula) because I don't want my teacher to think that I just copied and pasted stuff. I've spent too much time working on this for that to happen, so if she asks me I want to be able to explain the whole thing to her.

4. I think the paper was just trying to prove a point. 24 individuals in unique pairs is 276, and 100 individuals in unique pairs is 4950 - but they rounded up for effect.

The approximation $e^{-\frac{m^2}{2N}}=P$ is used to approximate the probablity that there will be NO matching birthdays. By using its compliment, you can find the probability approximation of finding at least one pair. That just for probabilities. If you want to find the value of a particular variable, in this case P, then yes you need to solve for "m", taking N to be fixed by the nature of your experiment.

I have no idea what an IB Math Project is, but I assume its important. If this is going to be your project, and it looks like an interesting one, then you will need to do some field research. What I would suggest, asking permission from some math instructors to get the birthdays of folks in their class. You might want to stay with the "smart kids" class, since if people aren't giving accurate answers, it can screw your data up. If you just turn in a paper with numbers you made up, then there really is no learning in action right? Nothing on that paper requires you to know anything other than what a sample space is, what the compliment of an event is, and how to do simple algebra.

And its not a paradox. I was expecting something sexy and exciting when you said paradox.

5. Oh yeah I've heard before that it isn't a paradox...It's just its common name. Sorry it wasn't really sexy, but I appreciate your help!

6. Originally Posted by ANDS!
And its not a paradox. I was expecting something sexy and exciting when you said paradox.
What is a paradox? It is a phenomenon that is surprising and counterintuitive at first sight. Once it is explained and understood, the paradox vanishes.

The so-called "birthday paradox" illustrates how people usually have defective intuitions about probabilities: it usually seems that a common birthday among 24 people is pretty unlikely (considering 24 is much less than 365). Of course, after some thought, we realize that it is not 24 that must be compared to 365, but the number of pairs, which is much larger, and of the same order of magnitude as 365, thus resolving the paradox.

If you weren't surprised to learn that among 24 people, a common birthday happens with probability more than 0.5, then it is not a paradox to you. It comes as a surprise to many people however, hence the name "paradox".

7. That wouldn't be a paradox to me - simply a surprising result. A paradox to me is something whose existence contradicts itself.

Lost in translation perhaps.

8. Originally Posted by ANDS!
This seems to be the definition of a paradox in a context of logic or philosophy. Quoting a dictionary,

1. a seemingly absurd or self-contradictory statement that is or may be true religious truths are often expressed in paradox
2. (Philosophy / Logic) a self-contradictory proposition, such as I always tell lies
3. a person or thing exhibiting apparently contradictory characteristics
4. an opinion that conflicts with common belief

9. This webpage may be useful here.
Quine wrote a good deal on this subject.