The following tables shows the house edge of most
casino games. For games partially of skill perfect play is assumed.
See
below the table for a definition of
the house edge.
| Game |
Bet/Rules |
House Edge |
Standard
Deviation |
| Baccarat |
Banker |
1.06% |
0.93 |
| Baccarat |
Player |
1.24% |
0.95 |
| Baccarat |
Tie |
14.36% |
2.64 |
| Big Six |
$1 |
11.11% |
0.99 |
| Big Six |
$2 |
16.67% |
1.34 |
| Big Six |
$5 |
22.22% |
2.02 |
| Big Six |
$10 |
18.52% |
2.88 |
| Big Six |
$20 |
22.22% |
3.97 |
| Big Six |
Joker/Logo |
24.07% |
5.35 |
| Bonus Six |
No insurance |
10.42% |
5.79 |
| Bonus Six |
With insurance |
23.83% |
6.51 |
| Blackjacka |
Atlantic City rules |
0.43% |
1.2 |
| Blackjackb |
Las Vegas single deck |
0.18% |
1.2 |
| Caribbean Stud Poker |
5.22% |
2.24 |
| Casino War |
Go to war on ties |
2.88% |
1.05 |
| Casino War |
Surrender on ties |
3.70% |
0.94 |
| Casino War |
Bet on tie |
18.65% |
8.32 |
| Catch a Wave |
0.50% |
d |
| Craps |
Pass/Come |
1.41% |
1.00 |
| Craps |
Don't pass/don't come |
1.36% |
0.99 |
| Craps |
Field (2:1 on 12) |
5.56% |
1.08 |
| Craps |
Field (3:1 on 12) |
2.78% |
1.14 |
| Craps |
Any craps |
11.11% |
2.51 |
| Craps |
Big 6,8 |
9.09% |
1.00 |
| Craps |
Hard 4,10 |
11.11% |
2.51 |
| Craps |
Hard 6,8 |
9.09% |
2.87 |
| Craps |
Place 6,8 |
1.52% |
1.08 |
| Craps |
Place 5,9 |
4.00% |
1.18 |
| Craps |
Place 4,10 |
6.67% |
1.32 |
| Craps |
Place (to lose) 4,10 |
3.03% |
0.69 |
| Craps |
Proposition 2,12 |
13.89% |
5.09 |
| Craps |
Proposition 3,11 |
11.11% |
3.66 |
| Craps |
Proposition 7 |
16.67% |
1.86 |
| Double Down Stud |
2.67% |
2.97 |
| Keno |
25%-29% |
1.30-46.04 |
| Let it Ride |
3.51% |
5.17 |
| Pai Gowc |
1.50% |
d |
| Pai Gow Pokerc |
1.46% |
0.75 |
| Red Dog |
Six decks |
2.80% |
d |
| Roulette (single zero) |
2.70% |
e |
| Roulette (double zero) |
5.26% |
e |
| Sic-Bo |
2.78%-33.33% |
e |
| Slot Machines |
2%-15%f |
8.74g |
| Spanish 21 |
Dealer hits soft 17 |
0.76% |
d |
| Spanish 21 |
Dealer stands on soft 17 |
0.40% |
d |
| Super Fun 21 |
0.94% |
d |
| Three Card Poker |
Pairplus |
2.32% |
2.91 |
| Three Card Poker |
Ante & play |
3.37% |
d |
| Video Poker |
Jacks or better (full pay) |
0.46% |
4.42 |
| Wild Hold 'em Fold 'em |
6.86% |
d |
Notes:
- a
- Atlantic City rules are 8 decks, dealer stands on soft 17,
player may double on any two cards, player may double after splitting,
one card to split aces, no surrender.
- b
- Las Vegas single deck rules are dealer hits on soft 17, player
may double on any two cards, player may not double after splitting,
one card to split aces, no surrender.
- c
- Assuming player plays the house way, playing one on one against
dealer, and half of bets made are as banker.
- d
- Yet to be determined.
- e
- Standard deviation depends on bet made.
- f
- Slot machine range is based on available returns from a major
manufacturer
- g
- Slot machine standard deviation based on just one machine.
While this can vary, the standard deviation on slot machines
are very high.
House Edge
The house edge is defined as the ratio of the average loss to the
initial bet. The house edge is not the ratio of money lost
to total money wagered. In some games the beginning wager is not
necessarily the ending wager. For example in blackjack, let it ride,
and Caribbean stud poker, the player may increase their bet when
the odds favor doing so. In these cases the additional money wagered
is not figured into the denominator for the purpose of determining
the house edge, thus increasing the measure of risk.
The reason that the house edge is relative to the original wager, not the
average wager, is that it makes it easier for the player to estimate how much
they will lose. For example if a player knows the house edge in blackjack is
0.6% he can assume that for every $10 wager original wager he makes he will
lose 6 cents on the average. Most players are not going to know how much their
average wager will be in games like blackjack relative to the original wager,
thus any statistic based on the average wager would be difficult to apply to
real life questions.
The conventional definition can be helpful for players determine how much
it will cost them to play, given the information they already know. However
the statistic is very biased as a measure of risk. In Caribbean stud poker,
for example, the house edge is 5.22%, which is close to that of double zero
roulette at 5.26%. However the ratio of average money lost to average money
wagered in Carribean stud is only 2.56%. The player only looking at the house
edge may be indifferent between roulette and Caribbean stud poker, based only
the house edge. If one wants to compare one game against another I believe
it is better to look at the ratio of money lost to money wagered, which would
show Caribbean stud poker to be a much better gamble than roulette.
Many other sources do not count ties in the house edge calculation, especially
for the don't pass bet in craps and the banker and player bets in baccarat.
The rationale is that if a bet isn't resolved then it should be ignorred. I
personally opt to include ties although I respect the other definition.
Element of Risk
For purposes of comparing one game to another I would like to propose a different
measurement of
risk, which I call the "element of risk." This measurement is defined as the
average loss divided by total money bet. For bets in which the initial bet is
always the final bet there would be no difference between this statistic and
the house edge. Bets in which there is a
difference are listed below.
| Game |
Bet |
House Edge |
Element
of Risk |
| Blackjack |
Atlantic City rules |
0.43% |
0.38% |
| Bonus 6 |
No insurance |
10.42% |
5.41% |
| Bonus 6 |
With insurance |
23.83% |
6.42% |
| Caribbean Stud Poker |
5.22% |
2.56% |
| Casino War |
Go to war on ties |
2.88% |
2.68% |
| Double Down Stud |
2.67% |
2.13% |
| Let it Ride |
3.51% |
2.85% |
| Spanish 21 |
Dealer hits soft 17 |
0.76% |
0.65% |
| Spanish 21 |
Dealer stands on soft 17 |
0.40% |
0.30% |
| Three Card Poker |
Ante & play |
3.37% |
2.01% |
| Wild Hold 'em Fold 'em |
6.86% |
3.23% |
Standard Deviation
The standard deviation is a measure of how volatile your bankroll will be playing
a given game. This statistic is commonly used to calculate the probability that
the end result of a session of a defined number of bets will be within certain
bounds.
The standard deviation of the final result over n bets is the product of
the standard deviation for one bet (see table) and the square root of the number
of initial bets made in the session. This assumes that all bets made are of
equal size. The probability that the session outcome will be within one standard
deviation is 68.26%. The probability that the session outcome will be within
two standard deviations is 95.46%. The probability that the session outcome
will be within three standard deviations is 99.74%. The following table shows
the probability that a session outcome will come within various numbers of
standard deviations.
| Number |
Probability |
| 0.25 |
0.1974 |
| 0.50 |
0.3830 |
| 0.75 |
0.5468 |
| 1.00 |
0.6826 |
| 1.25 |
0.7888 |
| 1.50 |
0.8664 |
| 1.75 |
0.9198 |
| 2.00 |
0.9546 |
| 2.25 |
0.9756 |
| 2.50 |
0.9876 |
| 2.75 |
0.9940 |
| 3.00 |
0.9974 |
| 3.25 |
0.9988 |
| 3.50 |
0.9996 |
| 3.75 |
0.9998 |
I realize that this explanation may not make much sense to someone who is
not well versed in the basics of statistics. If this is the case I would recommend
enriching yourself with a good introductory statistics book. There is also
a good definition of the term and examples here.
Hold
Although I do not mention hold percentages on my site the term is worth defining
because it comes up a lot. The hold percentage is the ratio of chips the casino
keeps to the total chips sold. This is generally measured over an entire shift.
For example if blackjack table x takes in $1000 in the drop box and of the $1000
in chips sold the table keeps $300 of them (players walked away with the other
$700) then the game's hold is 30%. If every player loses their entire purchase
of chips then the hold will be 100%. It is possible for the hold to exceed 100%
if players carry to the table chips purchased at another table. A mathematician
alone can not determine the hold because it depends on how long the player will
sit at the table and the same money circulates back and forth. There is a lot
of confusion between the house edge and hold, especially among casino personnel.
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Better known as "The Wizard of Odds", Michael Shackleford
uses math and computer analysis to determine optimal playing strategy for
all casino games. His work can be found on his website, WizardofOdds.com
with player strategies and probabilities on most of the casino games. Michael
lives in Las Vegas and in his spare time likes to college license plates
and gamble.
You can read more about Michael's work at his website, WizardofOdds.com |
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