The Expected Value of Pyramid Dice (Casino Game) #SoME2
Channel: Kevin Lubick
Date: 2022-07-16

A video that shows how to calculate the Expected Value of the Casino Game Pyramid Dice. This is an interesting, but not too complex combinatorics problem. 

Prior to this video (July 2022), there was no complete and publicly available analysis of Pyramid Dice. The analysis in this video is original research by Kaylee Lubick and shows Pyramid Dice has an Expected Value of -12.37% as originally designed.

Google Sheet with formula implemented and simulation results:
 - https://bit.ly/PyramidDice

Source Code for simulation:
 - https://gist.github.com/kjlubick/6c35ba0dfb80b3df8ffa666fb2ec76ca

Corrections
 - Starting at 11:25 and again at 19:14,  Reroll(d,n) should be d+2n not d*2n. (Thankfully, the calculations in the spreadsheet part use the correct formula, so the final results are correct) Reported by britboy3456

Mathematics of Casino Carnival Games by Mark Bollman
 - https://www.barnesandnoble.com/w/mathematics-of-casino-carnival-games-mark-bollman/1136896222

House Edge of Casino Games
 - https://wizardofodds.com/gambling/house-edge/

Mathologer Video on making change for 1 googol dollars
 - https://www.youtube.com/watch?v=VLbePGBOVeg

Standup Maths video on infinitely fragile primes.
 - https://www.youtube.com/watch?v=p3Khnx0lUDE&t=1395s
 - We can prove they exist, but cannot calculate them

Animations created by Kaylee Lubick with https://www.manim.community/

Casino Game Operator voiced by Bob Petts. Other Narration by James K.

This was an entry into Summer of Math Exposition 2 #SoME2
 - https://www.reddit.com/r/math/comments/v89f9q/summer_of_math_exposition_2/

0:00 - Intro
0:30 - Rules of Pyramid Dice
0:54 - Example Play 1 (Win)
1:37 - Example Play 2 (Lose)
2:00 - Example Play 3 (Push)
2:42 - Expected Value - How Rigged are Casino Games like Roulette?
4:02 - Overview of Calculation Process
4:23 - Setting up Calculation Spreadsheet
5:40 - Working out Small Cases
7:30 - Defining the Bust function
11:00 - Defining the Reroll function
11:40 - Combinations (P choose D) arrive on the scene
13:26 - Implementing the Closed-form Formula in a Spreadsheet
16:14 - Validating the Formula using Simulation
17:23 - Probability Distribution calculated to at least 5 significant figures
17:40 - EV calculations of the base game and some variants
18:54 - Recap and Outro