Although people have played casino games like Roulette for centuries, some of the elementary facts about betting systems are unknown. For example, most people will insist that your chances at Keno are much much worse than your chances betting a color at the roulette wheel. As we will see, this isn't completely true. I couldn't find the information shown on this page anywhere by Googling, so I wrote this page myself!

Let's start with some elementary definitions about a casino bet, which we call "bet A".

- p
_{A}= the probability you will win the bet. - w
_{A}= the amount returned by the casino if you bet $1 and win. - v
_{A}= 1 - p_{A}·w_{A}= the House commission

Here the payoff is called "*w _{A} for 1*."
It can also be called "

For example, if you bet a single number on an American Roulette wheel:

- p
_{A}= 1/38 = .026316 - w
_{A}= 36 - v
_{A}= 1 - p_{A}·w_{A}= 1 - 36/38 = .052632

- p
_{B}= 18/38 = .473684 - w
_{B}= 2 - v
_{B}= 1 - p_{B}·w_{B}= 1 - 36/38 = .052632

The House vigorish here is greater than zero, and
we will assume v_{A} > 0 for the remainder of this webpage.
Some people think that since v_{A} is constant at the
roulette wheel, it doesn't matter how you bet: you could
sprinkle your money on the table randomly.
This is not true however.
As an extreme example, suppose you bet 1/38 of your stake on *each*
of the 38 numbers!
The combined effect of these 38 bets is a single bet "C" with the
following parameters:

- p
_{C}= 38/38 = 1 - w
_{C}= 36/38 = .947368 - v
_{C}= 1 - p_{C}·w_{C}= 1 - 36/38 = .052632

This section is so basic, I'm afraid many readers have gotten
bored already! But things will get more interesting.
We will introduce the idea of fashioning your own bet from two
or more spins of the roulette wheel.
Throughout the remaining discussion we will assume that casino
allows you to bet *any* non-negative amount of money you want
to whether it be $1,000,000 or 39.3 pennies.
(This assumption simplifies the math and the discussion,
even though it may not correspond with real casinos.)
The only restriction is that the casino will not offer credit:
You must have a bankroll large enough to actually place the wager.

*Please note that throughout this page
we consider only the refashioning of a single bet*.
The simplest way to focus on this assumption is to suppose
you have $Z

Without loss of generality, we may assume Z_{start} = 1.

Let's start with a very simple example. You want to quadruple your money at the roulette wheel. Your choices include betting on a group of nine numbers (e.g. 1 to 9), or betting on a color and, if you win, letting your chips all ride one more time on that same color. (As we will see later, there are better options, but let's start here.)

We already know the house vigorish on the first option; it's .052632. Let's look at the second option (back-to-back bets on a color):

- p
_{T}= 18/38 · 18/38 = .224377 - w
_{T}= 2 · 2 = 4 - v
_{T}= 1 - p_{T}·w_{T}= 1 - (36/38 · 36/38) = .102493

Anyway, what we have done with this example is
to fashion a compound bet "T" from a simple bet "A"
of parameters w_{A} and v_{A}.
The compound bet we fashion can have whatever w_{T}
we choose it to have.
Because you are now fashioning *your own* bet, with
*your own* goal target, rather than directly applying
a bet the casino offers, we will call this bet "T",
"T" for *target*.
The question is: What is the efficacy of the compound bet?

The remainder of this page considers just one question:
how do we construct the compound bet "T" to minimize v_{T}
and what is that minimal v_{T}?

We will derive this optimal v_{T} as a function
of w_{A}, v_{A} and w_{T}.

If you want to increase your $1 to $1024 and the casino offers you no choice but an even-money bet, your strategy should be obvious: Bet everything, and continue to let it ride, hoping that you win ten times in a row.

More generally, when

w_{T} = (w_{A})^{K}

for a positive integer K, you let your money ride K times.
Since

p_{A} = (v_{A} - 1) / w_{A}

it follows that

p_{T}
= (v_{A} - 1)^{K} / w_{A}^{K}

which leads to

v_{T} = 1 - (v_{A} - 1)^{K}

or

v_{T} = 1 - (v_{A} - 1)^{
(log wT / log wA)}

Proving that "Bet Entire Bankroll" is the correct strategy in this case
is straightforward: it minimizes the total action you expose to the House
vigorish.
The final equation will be exactly valid only when
K = *log* w_{T} / *log* w_{A}
is a positive integer.

If you have $1023 but need $1024, and your only opportunity
is a casino which offers only
an even-money bet, your best strategy is to bet $1 and
of course walk away if you win.
If you lose, bet $2; losing again, bet $4 and so on.
This is the famous *Martingale* but please think carefully
before laughing!
I'm not saying the *Martingale* is a casino-beating idea,
just that it's the best strategy (and indeed obviously so)
*with the stated condition*: That you need $1024 and don't
care if, when unsuccessful, you lose your entire $1023 savings.
As before, we have fashioned our own bet "T" from casino's bet "A."
In this case, w_{A} = 2 and w_{T} = 1024/1023.

More generally, you will bet G_{1}, G_{2},
G_{3}, G_{4}, ...
where

G_{s} = (w_{T} - 1) ·
*Sum*_{(k=1,2,...,s)} (w_{A} - 1)^{-k}
= (w_{T} - 1) ·
w_{A}^{s-1} / (w_{A} - 1)^{s}

You will have the opportunity to make exactly M bets when

1 = G_{1} + G_{2} + ... + G_{M}
= (w_{T} - 1)
· ((w_{A} / (w_{A} - 1))^{M} - 1)

When we solve for M, we get something complicated, but that
matters little as we intend just to plug this into computer programs.
The formula for M is:

M = *log*
(w_{T} / (w_{T} - 1))
/ *log* (w_{A} / (w_{A} - 1))

(For those following along carefully, in our very simple example
w_{A} = 2, w_{T} = 1024/1023 and this equation does
indeed yield M = 10 as expected.)

The probability of success on our compound bet "T" is

p_{T} = 1 - (1 - p_{A})^{M}
= 1 - (w_{A} + v_{A} - 1)^{M} · w_{A}^{-M}

so the net vigorish is

v_{T} = 1 - w_{T} + w_{T}
· (1 - p_{A})^{M}

or

v_{T} =
1 - w_{T} + w_{T} · (w_{A}
+ v_{A} - 1)^{M} · w_{A}^{-M}

Since M was derived above as a function of w_{T} and w_{A},
once again we've derived v_{T} as a function of the three
parameters w_{A}, v_{A} and w_{T}.
And once again, the formula is valid only when M is a positive integer.

In general, we should bet precisely what we need to achieve our
goal. These bets are the so-called *Martingale* bets
G_{1}, G_{2}, G_{3}, G_{4} ...
we saw in the last section.
The exception is when that bet would exceed our total bankroll;
then we bet our entire bankroll instead.
The analysis of efficacy (which can be denoted by the effective
vigorish v_{T}) thus uses both the formulae

v_{T} = 1 - (v_{A} - 1)^{
(log wT / log wA)}

and

v_{T} =
1 - w_{T} + w_{T}(w_{A}
+ v_{A} - 1)^{M} · w_{A}^{-M}

But these formulae are almost absurdly different!
I said I was going to give you a general formula, but I lied!
What I have is a C program you can compile and run
to find v_{T} for any input values
of w_{A}, v_{A} and w_{T}.
That program will also print out the vigorishes estimated by
each of our two formula.
It displays "need" (the number of successive winning bets needed in the
first formula) and "opps" (the number of winning opportunities
in the second formula).
You will see that the first vigorish estimate is fairly close to
the actual effective vigorish when "need" is large,
the second estimate close when "opps" is large.
(An estimate will be *exact*
only when one of these numbers is an integer.)

Given that you've restricted your options to a single
bet "A", with associated parameters, is it always optimal
to follow the advice in this section?
I believe so, though cannot show a rigorous proof.
It will sometimes be the case that an alternate betting
regime yields a success rate *identical*
to the described regime, but I don't think it can surpass it.

Throughout this page we've assumed the casino offers only
one bet, the one described by w_{A}, v_{A}.
How should you choose among choices when the casino
offers a variety of bets? (Say bets "A" and "B.")

- If w
_{A}≥ w_{B}and v_{A}≤ v_{B},

then bet "A" is almost always at least as good as bet "B". - There are some other clear cases, but often the best choice
*will depend on your goal*w_{T};

then use the program.

There are exceptions where the lower-payoff bet of equal
vigorish outperforms the higher-payoff bet slightly.
This usually arises when w_{A} is very close to your
target w_{T}; even then bet "B" is better when
w_{B} is much larger than w_{A}.
To give just one other example, when w_{T} = 8/7,
a w_{A} = 2 bet may outperform 2 < w_{B} < 2.4,
since it gives you an exact number (M = 3) of Martingale chances.

Instead of compiling and running the C program, you can use the Javascript form above, though it doesn't print as much information as the C program. The defaults are for doubling your money at roulette. (Sorry if it works poorly; this is my first real Javascript program!)

While passing time watching a roulette wheel, a millionaire, just for fun,
places $9,000 on Red and says "It's all your sif it wins!" (He then walks away.)
You realize this "ticket" is worth $8526 on average, but you'll probably finish with zero.
So you decide to place bets of your own: $9000 on Black and $500 each
on 0 and 00.
You've just laid out $10,000 but whatever happens the croupier will be shoving
$18,000 your way. A sure $8000 seems better than the highly uncertain $8526 "ticket."
But by the Kelly Criterion, whether you take this hedge is a function of your bankroll
B. You want to maximize the weighted *geometric* mean of (B + r), where
r is your win or loss on this bet.
Instead of $10,000 what's the total (X) you should bet on Not-Red to maximize that
weighted geometric mean?
Dust off your calculus and see if you get the same answer as I do.

Solution: X = ($144,000 - B) / 15.2 (Or zero, if your bankroll exceeds $144,000 and the casino doesn't allow negative bets!)

Copyright © 2010 by James D. Allen

Please send me some e-mail.