# Lucky Nines

## Introduction

I first noticed the Lucky Nines side bet on a baccarat machine on a bar top multi-game machine by Interblock at Binion's Las Vegas on October 13, 2019. I would later see it on June 26, 2023 at the Red Rock casino in Las Vegas with a more liberal pay table. The bet pays according to the number of nines between the Player's and Banker's initial two cards and whether they are suited.

## Rules

1. Eight decks are used.
2. Two cards are dealt to the Player and Banker hands each.
3. The Lucky Nines bet pays according to the following pay table and those initial four cards. Pays are on a "to one" basis.

### Lucky Nines Pay Table

Event Binion's
Pay Table
Red Rock
Pay Table
4 Nines 500 1,000
3 Nines, same suit 250 500
3 Nines, mixed suits 20 35
2 Nines, same suit 8 16
2 Nines, mixed suits 5 5
1 Nine of diamonds 2 2
1 Nine 1 1

## Analysis

The following table shows the number of combinations, probability, and contribution to the return of all possible outcomes under the Binion's pay table. The lower right cell shows a house edge of 19.46%.

### Binion's Return Table

Event Pays Combinations Probability Return
Four nines 500 35,960 0.000029 0.014619
Three suited nines 250 64,512 0.000052 0.013113
Three nines 20 1,840,128 0.001496 0.029922
Two suited nines 8 8,236,032 0.006696 0.053571
Two nines 5 28,237,824 0.022959 0.114794
One nine of diamonds 2 74,908,672 0.060905 0.121810
One nine 1 224,726,016 0.182714 0.182714
No nines -1 891,881,376 0.725148 -0.725148
Total   1,229,930,520 1.000000 -0.194605

The next table shows my analysis of the Lucky Nines under the Red Rock pay table. The lower right cell shows a house edge of 8.27%.

### Red Rock Return Table

Event Pays Combinations Probability Return
4 Nines 1000 35,960 0.000029 0.029237
3 Nines, same suit 500 86,016 0.000070 0.034968
3 Nines, mixed suits 35 1,818,624 0.001479 0.051752
2 Nines, same suit 16 8,236,032 0.006696 0.107141
2 Nines, mixed suits 5 28,237,824 0.022959 0.114794
1 Nine of diamonds 2 74,908,672 0.060905 0.121810
1 Nine 1 224,726,016 0.182714 0.182714
All other -1 891,881,376 0.725148 -0.725148
Total   1,229,930,520 1.000000 -0.082730

## Card Counting the Lucky Nines

The unit I saw this side bet on shuffled the cards between hands. If you find this side bet based on a live dealer game, it would be very vulnerable to card counting (shut up Wiz!).