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The Birthday Problem & Other Paradoxes: A Step-by-Step Guide [Intro to Conditional Probability]

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Tags: probabilitystatisticsparadox

A guide to the birthday problem (aka birthday paradox), the two child paradox and the rare disease testing paradox explained from first principles (without using Bayes' rule). This video introduces conditional probability and probability trees to explain the solution to these 3 classic probability paradoxes. *How the video fits the useful-for-teachers theme:* The video is intended to be used as an aid for advanced high school probability classes (e.g. AP statistics) or early undergraduate classes (e.g. intro stats or intro probability) to help teach conditional probability, which is often a hard to grasp topic for students. The problems here should be useful when just starting out, to introduce some interesting counterintuitive problems you can solve with these methods as motivation for the topic. Each of the 3 problems in the topic can be used as a standalone example problem, for example at the beginning of a class. I've also tried not to overload the video with too many new ideas: I've intentionally kept the notation lightweight so that formalized notation can be built on the main ideas afterwards. After this video, the groundwork for Bayes theorem should be nicely set . The treatment of the birthday paradox as asking "how many people?" also sets up the idea of random variables as a future topic too. See description in YouTube video for some more links to related content.

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6.98 Overall score*

27 Rank

14 Votes

7 Comments

Comments

I’d seen all of these “paradoxes” before, except the last one about the boy born on Tuesday. This made me feel the same way I felt back in school, that surely this cannot be true and it has to be some sort of clever accounting trick because it intuitively makes no sense. I’ve thought about the intuition more and even still I find it genuinely surprising. Almost makes me feel that that maths is wrong that it could lead to such a counterintuitive result

The clarity provided is really good and commendable but the questions tackled were really the YT probability -101 puzzles, right up there with the money hall problem. Some input or exploration as to how we can use it in today’s technology to remove bias and keep updating our knowledge system.

5.9

The explanations are good, but the video seems to be too long, and the problems are not kind of very related. I would suggest to have different videos for every problem instead. If one is not hooked from the beginning, chances are they will not finish the video.

6.5

The video is well explained and each of the three problems is interesting. However, at 50 minutes, it feels too long for a high school audience. Focusing on just one problem would have allowed for a deeper explanation of the basic or preliminary concepts, which are often assumed but not fully clarified.

Dynamic, pleasant, full of energy : that are actual pedagogic qualities.

I think the best part is the two child paradox, but the two others are nice too. This video can definitely be shown to high school students and will help them.

8.2

I loved this video! I had heard of (almost) all of these paradoxes before but I have never seen them all packaged in one video. And thinking back on it, this video should exist. The world needed a video where all three of these were together so students could more than just understand there intuition fails them in this one edge case, but get a better feel for the pattern in this whole class of problems where intuition fails us and hopefully through reckoning with that get a better understanding of how probability actually works. More than that, I thought your presentation was insanely easy to follow and accessible. I really enjoyed that you kept it fairly high level and did not get lost in the sauce of all three algebra. The higher level concepts here have really not much to do with algebra, so why confuse students by shoe horning in algebra in where it does not have to be? If I were teaching students probability using this video I would use this as a first inspiration and then in class and in homework have them do the more complicated algebra heavy examples. That’s what actual class is for I think not YouTube.

The only thing I wish I saw is more explanation of why our intuition fails us overall. You do give good explanations for why each individual problem fails to follow our expectation but I think a great value of this video is the juxtaposition of all of these problems together. So it would have been nice to see the explanations juxtaposed as well to try and uncover a general pattern of where our misconceptions lie. From my viewpoint it actually seems to be an issue of how bad our brains are at intuitively compremhending the amount of combinations that are possible.

But as I said, overall great video! I do not think it is for a student with no probability background, but for a students that knows all the basic principles this would be a great video for them to see. And even for myself I had never before heard of the Tuesday addition to the boy-girl paradox and now I’m going to be thinking about how that makes sense intuitively for a while!

Very nice lecture. It was great to see the two different approaches . However, the content has been around on youtube on various places.