Heads-Tails and Heads-Heads are different The ABRACADABRA Theorem for coin flips and dice #SoME2

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 for contest announcement! Related Links: (probability_theory) (betting_system) Blog post with more accurate histogram of HT vs HH: Blog posts on ABRACADABRA Theorem: Video with more detailed proof: Solutions using the Markov chain method shown in the intuition section: MindYourDecisions Numberphile Link to github that has Manim code in the video (warning: the code is truly horrible! read at your own peril!) Music by Vincent Rubinetti Thank you Vincent for giving permission to use your wonderful music! Download the music on Bandcamp: Stream the music on Spotify: Track used: Euler's Clock #maths #probability 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