home
 betting guide 
d'alembert system 
D'Alerbert System 

Another popular mathematical
system is named after Jean
Le Rond d'Alembert, a French
mathematician and physicist
who was born in 1717. His
theory on the "Law of
Equilibrium" supposes a
balance of successes and
failures of certain events
if you consider a long
series of these events. The
d'Alembert, sometimes
referred to as the "Pyramid
System," has you increase
your bet by one unit after a
loss and decrease your bet
by one unit after a win. The
sequence and amount raised
or lowered can be varied to
suit particular games and
odds.
Each loss go up one unit,
each win go down one unit.
Thus “hopefully” you get
back where you started,
meanwhile in the end making
$1 from each and every win.
Example:
1. First bet $1. If you win,
bet $1. If you lose, bet $2.
2. If you win $2, next bet
$1. If you lose, bet $3.
3. If you win $3, next bet
$2. If you lose, bet $4.
4. If you win $4, next bet
$3. If you lose, bet $5.
5. If you win $5, next bet
$4. If you lose, bet $6.
6. If you win $6, next bet
$5. If you lose, bet $8.
7. Etc.
You’d better set limits or
will find yourself in outer
space some daybut usually,
D’Alembert is the most
efficient system. For
example if you start at $1
and “return to base” every
100 plays, whether winning
or losingthis is definitely
better than flat betting at
$50.
One typical sequence would
be handled as follows:
1 bet) Bet 1 unit
Lose,
1 units
2 bet) Up to 2 units
Win,
+1 unit
3 bet) Bet 1 unit
Lose,
+0 units
4 bet) Up to 2 units
Lose,
2 units
5 bet) Up to 3 units
Win,
+1 unit
6 bet) Bet 2 units
Win,
+3 units
Your "unit" can be equal to
$1, $5, $25 or anything that
you designate. If your unit
were $5, then you would be
down $5 after the first
wager. Your second stake is
$10 and the win puts you up
to a net of one unit or $5.
Now you decrease your next
bet after a win, back to $5.
The loss of $5 puts you even
at zero units. The next bet
of two units loses so you
increase to three units.
Because you win this wager,
you will now decrease your
stake to two units. This
wager wins and you are up a
total of three units thus
far. There is no determined
stopwin point with the
system, so you must set one
for yourself. If one unit
profit were fine for you,
then you would have won the
sequence after the second
wager (being up one unit)
and quit or began a new
sequence. If two or three
units were your objective,
then the sixth bet would
have sufficed. The higher
your objective win, the
longer the sequence will be.
You should also preselect a
stoploss point for any
sequence that you play to
help control losses. Notice
that this sequence has three
wins and three losses. When
the wins and losses balance
each other, or are in
equilibrium, then your net
gain is equal to the number
of wins in your sequence.
This sequence has three wins
that balance out three
losses. The net gain is
three units.
Please realize that if we
had a losing sequence, a
more aggressive unit size
progression will work harder
against you, losing money
much faster. Because there
are more ways to lose than
win on an evenmoney wager
(18 wins versus 20 losses
out of 38 trials), you will
be on the losing side of the
sequence more often.
Therefore, you may choose to
portray a more favorable
sequence here as an example.
You are better, off in the
end, losing less with the
smaller unit size than
winning more with a larger
unit size. Let us examine
something called a "tree
diagram" of the d'Alembert
system. For this example, we
are using a $5 unit and will
limit the progression to no
more than five wagers:
The d´Alembert Tree Diagram
(5bet progression using a
$5 unit)
Total Probabilities of
sequence ending events =
1.00 or 100%
The tree diagram is called
that because it spreads out
as it grows, just as the
possibilities do. Starting
with one wager, you can
easily see how all the
possibilities develop going
up to five bets deep. Once
you know what all the
possible outcomes are, you
can calculate the likelihood
of each terminal event on
the tree. The terminal
events are represented with
rounded boxes and contain
the probability of reaching
that particular outcome. The
chances of winning the first
bet are easy to see. There
are 18 ways out of 38 to win
the wager; so, 18 divided by
38 equals 0.4737 or 47.37%.
In order to win after the
second bet you would have
lost the first, then won the
second. The chances of
losing the first wager
(20/38) times the chances of
winning the second (18/38)
are 24.93%. To calculate the
probability of reaching a
particular point on the tree
diagram, just count the
number of wins and loses
along the way and apply them
as exponents before
multiplying everything
together. We can calculate
the likelihood of winning a
sequence by losing three
bets and winning two bets,
for example, as in win #5:
P(Lose) x P(Lose) x P(Lose)
x P(Win) x P(Win) = P(Win
#5), which is the
probability that this exact
sequence will occur.
If: P(Win) = 18/38 and
P(Lose) = 20/38, for each
spin, then: (20/38)³ x
(18/38)² = P(Win #5).
P(Win #5) = 0.0327 or 3.27%
If you calculate all the
probabilities of terminal
events and add them
together, they should equal
1.00 (or 100%). A terminal
event is an event that
causes the progression to
end. A situation where the
bettor is ahead after the
first through fourth bets
would end the progression.
After placing the fifth
stake, win, lose or draw, we
have decided to quit the
sequence. Take the amount of
money that we are ahead or
behind for each terminal
event and multiple it times
the probability of that
event. Now sum these up to
calculate the average money
won or lost for this
particular betting system:
Win #1 ($5):
18/38 x $5
+$2.37
Win #2 ($5):
(20/38) x (18/38) x $5
+$1.25
Win #3 ($5):
(20/38)² x (18/38)² x $5
+$0.62
Win #4 ($5):
(20/38)³ x (18/38)² x $5
+$0.16
Win #5 ($5):
(20/38)³ x (18/38)² x $5
+$0.16
Average Total Winnings:
+$4.56
Lose ($25):
(20/38)^4 x (18/38) x –$25
–$0.91
Lose ($25):
(20/38)^4 x (18/38) x –$25
–$0.91
Lose ($25):
(20/38)^4 x (18/38) x –$25
–$0.91
Lose ($75):
(20/38)^5 x –$75
–$3.02
Average Total Loses:
<$5.75>
Allowing up to a 5bet
progression with $5 units,
the d'Alembert delivers
$4.56 in wins minus $5.75 in
loses, for a net loss of
$1.19 per betting sequence.
Another useful bit of
information is the average
number of spins, or bets per
progression. The summation
of the number of spins times
the probability of ending
the progression in as many
spins gives us this
statistic. For the first
four bets, the player must
win to end the sequence.
Otherwise, the sequence is
automatically terminated
after the fifth bet. You
will note, there is no
terminal event in the third
spin, so the probability of
ending the betting
progression is zero. Here is
how the calculation would
look:
Average number of spins for
a 5bet progression =
2.2951, or 2.3 spins.
We could have calculated the
probability of all six
terminal events in the fifth
spin and added them together
to get the probability of
going to five spins. Because
these events are mutually
exclusive, it is easier take
1.00 minus the chances of
ending the progression in
spins one through four. The
probability of ending in
spins one through four is
[0.4737 +0.2493 +0.0
+0.0622] or 0.7852.
Therefore, we have 100% –
78.52%, which equals a
21.48% chance of ending the
progression in the fifth
spin. Taking the sum of all
probabilities times the
spins needed is about 2.3
average spins per
progression for a 5bet
d'Alembert. If we lose $1.19
per progression and each
progression averages 2.3
spins, then we are expecting
a loss of almost 52 cents
per bet.










