Hard Rockin' Dice

Introduction

Hard Rockin' Dice offers three unique side bets, akin to a previous version known as 'Hot Hand', which rewards players if certain numbers appear before rolling a total of seven. This side bet was first introduced at the Jack casino in Cincinnati in March 2019 and was later renamed following the casino's transition to Hard Rock Cincinnati. Small, Tall, and All bets The Flaming Four wager rewards players with 70 to 1 odds if the shooter manages to roll a total of 2, 3, 11, and 12 before hitting seven.

Rules

1. With the Sizzling Six wager, players can earn 12 to 1 payouts if the shooter hits totals of 4, 5, 6, 8, 9, or 10 prior to rolling a seven.
2. The aim of the Hot Hand bet is to roll the numbers 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 before achieving a total of seven. Successfully doing so results in a payout of 80 to 1, while achieving 9 out of these 10 totals before hitting seven gives a payout of 20 to 1.
3. For further clarification, kindly refer to the official document.

The table below presents my analysis regarding the Flaming Four bet, where you can observe the house edge marked at 18.55% in the bottom right corner. rule card .

Analysis

In the table below, I've detailed my findings on the Sizzling Six bet, which reveals a house edge of 19.18% in the lower right cell.

Displayed below is my analysis of the Hot Hand bet. The house edge indicated in the bottom right corner is 18.02%.

Interestingly, this side bet can be analyzed using integral calculus. To determine the probability of winning outcomes, one must compute the integral from zero to infinity for the following formulas:

Methodology

f(x) = (1 - exp(-x/36))^2*(1 - exp(-x/18))^2*(1 - exp(-x/12))^2*(1 - exp(-x/9))^2*(1 - exp(-5x/36))^2*exp(-x/6)*(1/6)

• Totals of 2, 3, 11, and 12 rolled before a 7:
  f(x) = (1-exp(-x/36))^2*(1-exp(-x/18))^2*exp(-x/6)*(1/6)
  Integral = 53/4620 = apx. 0.01147186147186147
• Totals of 4, 5, 6, 8, 9, and 10 rolled before a 7:
  f(x) = (1-exp(-x/12))^2*(1-exp(-x/9))^2*(1-exp(-5x/36))^2*exp(-x/6)*(1/6)
  Integral = 832156379 / 13385572200 = Apx: 0.06216815886286878
• Totals of 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 rolled before a 7:
  The totals of 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 should be rolled before a 7, but missing 2 or 12:
  Integral = 126538525259/24067258815600 = Apx. 0.00525770409619644
• f(x) = (1 - exp(-x/36))*exp(-x/36)*(1 - exp(-x/18))^2*(1 - exp(-x/12))^2*(1 - exp(-x/9))^2*(1 - exp(-5x/36))^2*exp(-x/6)*(1/6)
  The outcomes of 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 must come before a 7, except for missing 3 or 11:
  Integral = 137124850157/24067258815600 = apx. 0.00569756826930859
• f(x) = (1 - exp(-x/36))^2*(1 - exp(-x/18))*exp(-x/18)*(1 - exp(-x/12))^2*(1 - exp(-x/9))^2*(1 - exp(-5x/36))^2*exp(-x/6)*(1/6)
  When rolling the totals of 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 before obtaining a 7, with the exception of missing 4 or 10:
  Integral = 150695431/75445952400 = apx. 0.001997395833788958
• f(x) = (1 - exp(-x/36))^2*(1 - exp(-x/18))^2*(1 - exp(-x/12))*exp(-x/12)*(1 - exp(-x/9))^2*(1 - exp(-5x/36))^2*exp(-x/6)*(1/6)
  For totals of 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12, they should be rolled before a 7, but missing 5 or 9:
  Integral = 1175248309/1266697832400 = apx. 0.000927804784171193
• f(x) = (1 - exp(-x/36))^2*(1 - exp(-x/18))^2*(1 - exp(-x/12))^2*(1 - exp(-x/9))*exp(-x/9)*(1 - exp(-5x/36))^2*exp(-x/6)*(1/6)
  The totals of 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 need to appear before a 7, except for missing 6 or 8:
  Integral = 35278/72747675 = apx. 0.0004849364601686583
• f(x) = f(x) = (1 - exp(-x/36))^2*(1 - exp(-x/18))^2*(1 - exp(-x/12))^2*(1 - exp(-x/9))^2*(1 - exp(-5x/36))*exp(-5x/36)*exp(-x/6)*(1/6)
  Correct mathematical techniques and insights for various casino games including blackjack, craps, roulette, among many others.
  Integral = 6534704369/24067258815600 = apx. 0.0002715184317029205

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Written by: Michael Shackleford