Last Updated: March 22, 2019

# Hot Hand

## Introduction

Hot Hand 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 game debuted at the Jack casino in Cincinnati in March 2019.

## 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.

## 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

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