Casino Over Under

Introduction

Casino Over Under is a simple game to learn and play. The object is to predict whether the total of three cards will be a low or high. The game made its debut at the Magnolia Bluffs casino in Natchez, Mississippi in April, 2016.

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Rules

Four to eight 52-card decks are used.
Cards are scored as in blackjack, except aces are always 11. In other words, 2-10 = pip value, face cards = 10 points, aces = 11 points.
Play starts with the player making an Ante and/or Bonus wager.
The dealer shall deal himself three cards, one face up and two face down.
The player has three choices:
- Make an Under bet.
- Make an Over bet.
- Fold
The Under or Over bet must be exactly equal to the Ante bet.
The dealer shall turn over the two face down cards and announce the total points.
If the player bet the Under and the total is under 17.5, then the player shall win even money on the Ante and Under bets. Otherwise, if the player made the Under bet, and the total is over 17.5, then both bets shall lose.
If the player bet the Over and the total is over 23.5, then the player shall win even money on the Ante and Over bets. Otherwise, if the player made the Over bet, and the total is under 23.5, then both bets shall lose.
The Bonus bet shall pay according to the three-card total and the pay table below.

The following pay table is for winning Bonus bets. All hands are based on three cards. All wins are on a "to one" basis.

Bonus Pay Table

Total Points	Pays
6 or 33	50 to 1
7 or 32	10 to 1
8 or 31	5 to 1
9 or 30	4 to 1
10 or 29	3 to 1
11 or 28	2 to 1
12 or 27	1 to 1
13 to 26	Loss

Strategy

With an up card of 5 or less, bet Under. Otherwise, if it is 6 or more, bet over. Never fold.

I would be remiss in my duties if I didn't at least mention that this game would be extremely vulnerable to card counting.

Analysis

Unless otherwise noted, all analysis on this page is based on six decks of cards.

Table 1 shows a summary of my analysis of the Ante bet. It shows the expected value of both the Under and Over bets for all 10 possible face-up cards. The probability column is the probability of being dealt that. The return column is the product of the probability and the greater of the two expected value columns. The bottom right column shows that for every unit bet on the Ante, the player can expect to lose 0.020997 units.

Table 1 — Ante Analysis by Face Up Card

Face Up Card	Expected Value Under	Expected Value Over	Probability	Return
2	0.262421	-1.977098	0.076923	0.020186
3	-0.072150	-1.785914	0.076923	-0.005550
4	-0.419666	-1.359734	0.076923	-0.032282
5	-0.798050	-1.120755	0.076923	-0.061388
6	-1.151623	-0.858874	0.076923	-0.066067
7	-1.338824	-0.582056	0.076923	-0.044774
8	-1.501131	-0.276278	0.076923	-0.021252
9	-1.644518	0.054310	0.076923	0.004178
10	-1.763012	0.407800	0.307692	0.125477
Ace	-1.858604	0.786184	0.076923	0.060476
Total			1.000000	-0.020997

Table 2 shows the probability and contribution to the return for all possible outcomes, assuming optimal strategy. As in table 1, the lower right cell shows the player can expect to lose 0.020997 per Ante bet. With a total wager of two units every time, the Element of Risk is 2.10%/2 = 1.05%.

Table 2 — Ante Bet Analysis by Outcome

Event	Pays	Combinations	Probability	Return
Win	2	7,441,032	0.494751	0.989502
Loss	-2	7,598,928	0.505249	-1.010498
Total		15,039,960	1.000000	-0.020997

Here is the house edge of the Ante bet for other numbers of decks:

4 decks: 2.3776%
8 decks: 1.9612%

Table 3 breaks down the possible outcomes of the Bonus bet. The lower right cell shows a house edge of 5.61%.

Table 3 — Bonus Bet Analysis

Event	Pays	Combinations	Probability	Return
6 or 33	50	4,048	0.000807	0.040372
7 or 32	10	33,120	0.006606	0.066064
8 or 31	5	129,312	0.025794	0.128968
9 or 30	4	227,272	0.045334	0.181335
10 or 29	3	211,680	0.042224	0.126671
11 or 28	2	259,200	0.051702	0.103405
12 or 27	1	312,496	0.062333	0.062333
13 to 26	-1	3,836,192	0.765200	-0.765200
Total		5,013,320	1.000000	-0.056052