search

leios / SoME_Topics

Collaboration / Topic requests for SoME
Other
212 stars 6 forks source link

A surprising fact about coinflips: It takes longer on average for HH to appear than TH! Why? #48

Open mcnica89 opened 2 years ago

mcnica89 commented 2 years ago

It takes on average 4 flips to see TH appear but 6 coinflips to see HH. How is this even possible? There is a truly creative proof of this fact! There is an old numberphile video about this problem (https://youtu.be/SDw2Pu0-H4g) and MindYourDecisions (https://www.youtube.com/watch?v=--mxW3jDlGk) but they do not show the really creative proof that works for any general sequence and do not give the general formula for any sequence)

About the author

I'm currently a professor at a university in Canada in the area of probability and machine learning. Last year I was a runner up in SoME1 for my video on the Buffon Noodle problem https://youtu.be/e-RUyCs9B08 . My websites where you can learn more about me: https://nicam.uoguelph.ca/ http://www.math.toronto.edu/mnica/

Quick Summary

Target audience: High school or early university students who have seen some random stuff and expected value.

First motivate the problem: how could it be that HH takes longer than TH? How do we calculate how long?

Then we derive the formula which tells you how long it will take for any sequence to appear: The explicit formula involves how many times substrings of the word appear in the word itself. The (completely counterintuitive) trick to derive the formula is to create a fair casino that has a gambling game where gambler's are using a particular betting strategy that terminates when the sequence appears. Because this a fair casio, there is a "no-free-lunch" theorem that states that the average money the casino makes is always 0. This can be exploited to get the formula! This formula is sometimes called the "ABRACADABRA Theorem"

(This "no-free-lunch" theorem is called Doob's optional stopping theorem, and the mathematical theorem being used is that of martingales...we could potnentially touch on this during the video!)

Target medium

Video! I am a manim beginner and could use some help from a manim expert (or someone who has lots of time to spend manim-ing!)

More details

The quick summary has the high level ideas...we could go into more or less depth about margingales and Doob's optional stopping time theorem (I think it would be best to keep it as understandable as possible so no fancy notation etc is needed) A detailed solution to the problem (which DOES use a lot of fancy math notation) can be found here https://elearning.unimib.it/pluginfile.php/583715/mod_resource/content/1/abracadabra.pdf

Contact details

Email: mcnica89@gmail.com

Additional context

Daniel-Groves commented 2 years ago

I left you an email

noamtashma commented 2 years ago

See Here for a few solutions

mcnica89 commented 2 years ago

See Here for a few solutions

The whole point is to do the fair casino proof which is much more elegant and really is (in my opinion) a beautiful application of using the no-free-lunch theorem to compute things! The proofs at this link are not nearly as elegant (and in fact the existence of this stack excahnge confirms to me this would be a good video!) The solution I want to do is outline at https://elearning.unimib.it/pluginfile.php/583715/mod_resource/content/1/abracadabra.pdf (Although again, the whole point is to make it accessible without having to define filtrations etc)

Daniel-Groves commented 2 years ago

@mcnica89 not to nag, but did you catch my email?

alan2here commented 2 years ago

The Pennies Game, and simplified versions of the Monty Hall problem. I love math but I cant help but consider this to be statistical mathematical black magic. I hope your video helps me finally understand it properly.