A Coin Flip Paradox and the ABRACADABRA Theorem for infinite monkeys: How long does it take?

Which takes more coin flips to come up on average, Heads-Tails or Heads-Heads? What about HHTT vs HTHT? Find out how long it takes and why using the magical ABRACADABRA Theorem. The method also works for dice rolls, the speeding rule in Monopoly, or a monkey typing at a typewriter.

Video created for the ‪@3blue1brown‬ Summer of Math Exposition 2 #SoME2
See    • Summer of Math Exposition 2 Invitation   for contest announcement!


Related Links:
https://en.wikipedia.org/wiki/Marting...)
https://en.wikipedia.org/wiki/Optiona...
https://en.wikipedia.org/wiki/St._Pet...
https://en.wikipedia.org/wiki/Marting...)


Blog post with more accurate histogram of HT vs HH:
https://blogs.sas.com/content/iml/201...


Blog posts on ABRACADABRA Theorem:
https://jeremykun.com/2014/03/03/mart...
https://martingalemeasure.wordpress.c...

Video with more detailed proof:
   • 11-07. Martingale theory - Optional s...  

Solutions using the Markov chain method shown in the intuition section:
MindYourDecisions    • Counter-Intuitive Probability. Coin F...  
Numberphile    • Consecutive Coin Flips - Numberphile  


Link to github that has Manim code in the video (warning: the code is truly horrible! read at your own peril!) https://github.com/mcnica89/manim/blo...


Music by Vincent Rubinetti
Thank you Vincent for giving permission to use your wonderful music!
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjw...
Track used: Euler's Clock

#math #maths #probability #3blue1brown #some2 #animation

Timestamps:
00:00 HH vs HT
00:36 Simulations and Histogram
02:13 Theoretical averages and E
02:54 HHTT vs HTHT is counterintuitive
03:50 Intuition by partial progress
05:43 Fair casinos and fair bets
06:19 The optional stopping theorem
07:41 Fair casino setup for HT
09:04 Example THHHT and E[N_HT] =4
11:57 Fair casino setup for HH
12:55 Why HH is different and E[N_HH] =6
14:12 Fair casino setup for HTHT
14:34 Why only the last flips matter
15:26 E[N_HTHT] =16+4
15:38 Fair casino setup for HHTT
15:12 E[N_HHTT] = 16
16:12 Other examples and E[N_HHH] =13
16:42 Number of dice rolls to get 666
17:18 Doubles three times in a row
17:39 Monkey at a typewriter
18:40 ABRACADABRA Theorem
19:11 Probability with Martingales