 Last Updated: March 22, 2019

# Hard Rockin' Dice

## Introduction

Hard Rockin' Dice is a set of three side bets, similar to the Small, Tall, and All bets, which win if a group of numbers are rolled before a total of seven. The side bet debuted at the Jack casino in Cincinnati in March 2019, which called it the "Hot Hand." When that casino changed hands to become the Hard Rock Cincinnati, the name of the side bet changed to Hard Rockin' Dice.

## Rules

1. The Flaming Four bet pays 70 to 1 if the shooter rolls a total of 2, 3, 11, and 12 before a total of seven.
2. The Sizzling Six bet pays 12 to 1 if the shooter rolls a total of 4, 5, 6, 8, 9, and 10 before a total of seven.
3. The object of the Hot Hand bet is to roll a 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 before a total of seven. If this is accomplished, then winning bets shall pay 80 to 1. If 9 out of 10 of these totals are achieved before a seven, then winning bets shall pay 20 to 1.

If that was not clear, please see the official rule card.

## Analysis

The following table shows my analysis of the Flaming Four bet. The lower right cell shows a house edge of 18.55%.

### Flaming Four

Event Pays Probability Return
Win 70 0.011472 0.803030
Lose -1 0.988528 -0.988528
Total   1.000000 -0.185498

The following table shows my analysis of the Sizzling Six bet. The lower right cell shows a house edge of 19.18%.

### Sizzling Six

Event Pays Probability Return
Win 12 0.062168 0.746022
Lose -1 0.937832 -0.937832
Total   1.000000 -0.191810

The following table shows my analysis of the Hot Hand bet. The lower right cell shows a house edge of 18.02%.

### Hot Hand

Event Pays Probability Return
10 80 0.005258 0.420616
9 20 0.018758 0.375169
0 to 8 -1 0.975984 -0.975984
Total   1.000000 -0.180199

## Methodology

This side bet can be, surprisingly, solved with integral calculus. To find the probability of all winning events, take the integral of 0 to infinity of the following functions:

• 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:
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))^2exp(-x/6)*(1/6)
Integral = 126538525259/24067258815600 = Apx. 0.00525770409619644
• Totals of 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 rolled before a 7, except missing 2 or 12:
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))^2exp(-x/6)*(1/6)
Integral = 137124850157/24067258815600 = apx. 0.00569756826930859
• Totals of 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 rolled before a 7, except missing 3 or 11:
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))^2exp(-x/6)*(1/6)
Integral = 150695431/75445952400 = apx. 0.001997395833788958
• Totals of 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 rolled before a 7, except missing 4 or 10:
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))^2exp(-x/6)*(1/6)
Integral = 1175248309/1266697832400 = apx. 0.000927804784171193
• Totals of 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 rolled before a 7, except missing 5 or 9:
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))^2exp(-x/6)*(1/6)
Integral = 35278/72747675 = apx. 0.0004849364601686583
• Totals of 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 rolled before a 7, except missing 6 or 8:
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)
Integral = 6534704369/24067258815600 = apx. 0.0002715184317029205

Suggested integral calculator

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