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Statistics 101 for Gamblers
is no need to be a mathematician to gamble in a casino, but a little
mathematical knowledge is all that is needed to make you a realistic,
if not a winning, gambler.
you gamble long enough in a casino, you will lose all of your money.
The simple reason is that the casino has constructed the games so that
they make money (on average) on every bet you make. For example, there
are 38 numbers on a roulette wheel, but the casino pays as if there were
only 36. In the long run, the casino keeps $2 of every $38 wagered at
double-zero roulette. There are a few expert players who can actually profit
in the long run in the casino, for exactly the reverse reason. An
expert video poker player playing full pay Deuces Wild will (over millions
of hands) win 7 cents of every $10 she wagers. Expert blackjack players
counting cards in a deeply dealt game, and disciplined players receiving
generous casino comps, can also profit in the long run. For online
players, the GameMaster's directory at http://www.gamemasterlist.com
will tell you the casino's advantage (or player's advantage) on most of the
games you will find online.
the "short run", however, all of these players can and will have winning and
losing nights, weeks, or even months. Gambling is a risky undertaking.
The casino may average a 5 cent profit on every $10 bet at blackjack,
but if the casino always won exactly 5 cents per hand, no one would play.
The risk inherent in gambling can be measured by a quantity called the
standard deviation. If your gambling results are "normally" distributed,
then 67% of the time your bankroll will end up within one standard deviation
above or below your average ("expected") result. Two standard deviations
contain 95% of our results, three standard deviations contain 99.7% of
all outcomes. The word "normal" here
is a technical word referring to the normal curve (bell curve) in statistics.
However, we can gloss over this technicality due to a
mathematical fact called the
"law of large numbers" which in our case says that if you make enough
wagers, your bankroll will follow a normal distribution. For games with a
relatively even distribution of outcomes (like blackjack) "enough" may be
just 100 bets, but for jackpot games with rare high payout events like
video poker, the long run can be quite long indeed. A rough rule of thumb
might require that you play long enough to hit the rare jackpot (on
average) 5-10 times to consider yourself in "the long run".
said that the GameMaster's directory listed the casino advantage for most
online games, but not the standard deviation. Fortunately, the standard
deviation is easy to calculate for most games. The variance of a game
is the average squared result of your wager. For craps with no odds, this
is trivial; you always win one bet or lose one bet, and the variance is 1.
If you make column bets in American roulette, in 38 spins you will win 2 bets
12 times, and lose one bet 26 times. The variance is
1/38 x (12 x 4 + 26 x 1) = 1.95. (To be precise, we should subtract off
the square of the casino advantage, but this correction should be negligible
if you are playing any reasonable game.) The standard deviation of the
game is simply the square root of the variance, for instance the standard
deviation of the column bet at roulette above is 1.4 bets.
can we do with these numbers? Assume that we take 100 spins betting $10 on
the first column at double zero roulette. On average, we expect to lose
5.26% of our $1000 in bets, or about $50. The standard deviation of our
100 spins is found by multiplying the game's standard deviation
(1.4 bets from above) by the square root of the number of spins (10)
to get a standard deviation of $140. Thus while our probable results will be
centered around $53, 95% of our results will fall between a win of $230 and
a loss of $330. If you lose more than $470 (three standard deviations below
the average), you have hit a horrible string of luck that should happen
only about 1 time in 1000, and you
may be justifiably concerned about the fairness
of the game.
your game is blackjack, the standard deviation per hand is roughly 1.1 bets.
The number is slightly larger than one because splits, doubles, and blackjacks
all give you the opportunity to win or lose more than one bet per hand.
Here is a recent example, unfortunately drawn from real life. I played
135 hands of blackjack at Casino Bar when they offered a 35% deposit bonus
to try their new software.
The rules here are horrendous (D10 only in multi-deck), and I will assume
a theoretical casino edge of 1%. Playing 135 hands, I expected to lose an
average of 1.4 bets. The square root of 135 times 1.1 gives a standard
deviation of 12.8 bets. A little more than one time in 1000 I could
expect to lose about 40 bets. My actual loss was 60 bets, or 4.6 standard
deviations below the expected result. This should happen only 5 times in
a million sessions in a fair game; for practical purposes, this is proof
that I was cheated by their new software, whether by bugs or intentionally.
that you know the casino advantage and standard deviation for your favorite
game, what should you do with these numbers? One thing for online players
is to monitor your actual results compared to your expected outcome. While
a casino could cheat in a subtle way and avoid detection, if you see results
from 100 or more hands that are outside of three standard deviations from
your expectation, you should be suspicious.
If other players you trust have reasonable results, it's possible that you
are indeed that one person in 1000 who will have such horrible luck, but
generally, these numbers can give you a realistic idea of the risks
inherent in gambling. If you like to play blackjack at $10 a hand, at
200 hands per hour for four hours at a time, your standard deviation
is about $300. You could easily be behind by $600 after a session, or
more if you are seriously unlucky. If that number gives you cause for
concern, you are betting too high. Of course, you could just as easily
win $600, and if you can accept this level of risk,
go ahead and place your bets, knowing you are gambling responsibly within
any comments or suggestions to me at JamesOnGambling@casino.com.
until next time, may your results exceed your expectations.
The GameMaster Online, Inc.
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