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Formulas and Results
To simplify things and also to be conservative I will assume you lose
50% of hands,
win 40% and have a stand off in 10%. Firstly let’s assume
there are no bonuses and you are playing with all your own money. You
have $300 and you want to wager a total of $1200. Also let’s assume you
will gamble the $300 each hand. The most hands you would play are 4. The
possible outcomes for each hand are:
S = Stand off (Probability = 0.1)
W = Win (Probability = 0.4)
L = Lose (Probability = 0.5)
The possible outcomes over 4 hands, and their probabilities are:
|
Outcomes
|
Probability (Pr)
|
Return (R)
|
Expected Return (Pr
X R)
|
|
L
|
0.5
|
-$300
|
-$150.00
|
|
WLL
|
(0.4x0.5x0.5) = 0.1
|
-$300
|
-$30.00
|
|
WLWL
|
(0.4x0.5x0.4x0.5) = 0.04
|
$0
|
$0.00
|
|
WLWW
|
(0.4x0.5x0.4x0.4) = 0.032
|
$600
|
$19.20
|
|
WLWS
|
(0.4x0.5x0.4x0.1) = 0.008
|
$300
|
$2.40
|
|
WLSL
|
(0.4x0.5x0.1x0.5) = 0.01
|
-$300
|
-$3.00
|
|
WLSW
|
(0.4x0.5x0.1x0.4) = 0.008
|
$300
|
$2.40
|
|
WLSS
|
(0.4x0.5x0.1x0.1) = 0.002
|
$0
|
$0.00
|
|
WWLL
|
(0.4x0.4x0.5x0.5) = 0.04
|
$0
|
$0.00
|
|
WWLW
|
(0.4x0.4x0.5x0.4) = 0.032
|
$600
|
$19.20
|
|
WWLS
|
(0.4x0.4x0.5x0.1) = 0.008
|
$300
|
$2.40
|
|
WWWL
|
(0.4x0.4x0.4x0.5) = 0.032
|
$600
|
$19.20
|
|
WWWW
|
(0.4x0.4x0.4x0.4) = 0.0256
|
$1200
|
$30.72
|
|
WWWS
|
(0.4x0.4x0.4x0.1) = 0.0064
|
$900
|
$5.76
|
|
WWSL
|
(0.4x0.4x0.1x0.5) = 0.008
|
$300
|
$2.40
|
|
WWSW
|
(0.4x0.4x0.1x0.4) = 0.0064
|
$900
|
$5.76
|
|
WWSS
|
(0.4x0.4x0.1x0.1) = 0.0016
|
$600
|
$0.96
|
|
WSLL
|
(0.4x0.1x0.5x0.5) = 0.01
|
-$300
|
-$3.00
|
|
WSLW
|
(0.4x0.1x0.5x0.4) = 0.008
|
$300
|
$2.40
|
|
WSLS
|
(0.4x0.1x0.5x0.1) = 0.002
|
$0
|
$0.00
|
|
WSWL
|
(0.4x0.1x0.4x0.5) = 0.008
|
$300
|
$2.40
|
|
WSWW
|
(0.4x0.1x0.4x0.4) = 0.0064
|
$900
|
$5.76
|
|
WSWS
|
(0.4x0.1x0.4x0.1) = 0.0016
|
$600
|
$0.96
|
|
WSSL
|
(0.4x0.1x0.1x0.5) = 0.002
|
$0
|
$0.00
|
|
WSSW
|
(0.4x0.1x0.1x0.4) = 0.0016
|
$600
|
$0.96
|
|
WSSS
|
(0.4x0.1x0.1x0.1) = 0.0004
|
$300
|
$0.12
|
|
SL
|
(0.1x0.5) = 0.05
|
-$300
|
-$15.00
|
|
SWLL
|
(0.1x0.4x0.5x0.5) = 0.01
|
-$300
|
-$3.00
|
|
SWLW
|
(0.1x0.4x0.5x0.4) = 0.008
|
$300
|
$2.40
|
|
SWLS
|
(0.1x0.4x0.5x0.1) = 0.002
|
$0
|
$0.00
|
|
SWWL
|
(0.1x0.4x0.4x0.5) = 0.008
|
$300
|
$2.40
|
|
SWWW
|
(0.1x0.4x0.4x0.4) = 0.0064
|
$900
|
$5.76
|
|
SWWS
|
(0.1x0.4x0.4x0.1) = 0.0016
|
$600
|
$0.96
|
|
SWSL
|
(0.1x0.4x0.1x0.5) = 0.002
|
$0
|
$0.00
|
|
SWSW
|
(0.1x0.4x0.1x0.4) = 0.0016
|
$600
|
$0.96
|
|
SWSS
|
(0.1x0.4x0.1x0.1) = 0.0004
|
$300
|
$0.12
|
|
SSL
|
(0.1x0.1x0.5) = 0.005
|
-$300
|
-$1.50
|
|
SSWL
|
(0.1x0.1x0.4x0.5) = 0.002
|
$0
|
$0.00
|
|
SSWW
|
(0.1x0.1x0.4x0.4) = 0.0016
|
$600
|
$0.96
|
|
SSWS
|
(0.1x0.1x0.4x0.1) = 0.0004
|
$300
|
$0.12
|
|
SSSL
|
(0.1x0.1x0.1x0.5) = 0.0005
|
-$300
|
-$0.15
|
|
SSSW
|
(0.1x0.1x0.1x0.4) = 0.0004
|
$300
|
$0.12
|
|
SSSS
|
(0.1x0.1x0.1x0.1) = 0.0001
|
$0
|
$0.00
|
|
Total
|
1
|
|
-$17.85
|
What the table
above shows is that (assuming Pr(W) = 0.4, Pr(L) = 0.5 and Pr(S) = 0.1)
if you gambled at an online casino with your own money and you made 4
bets of $300, on average you would expect to lose $17.85 or 5.95% of
your money.
Using the bonus money instead of your own, the winnings will not change
however whenever you lose you only lose $100 not $300. The table below
shows the expected results for 4 hands.
|
Outcomes
|
Probability (Pr)
|
Return (R)
|
Expected Return (Pr X R)
|
|
L
|
0.5
|
-$100
|
-$50.00
|
|
WLL
|
(0.4x0.5x0.5) = 0.1
|
-$100
|
-$10.00
|
|
WLWL
|
(0.4x0.5x0.4x0.5) = 0.04
|
$0
|
$0.00
|
|
WLWW
|
(0.4x0.5x0.4x0.4) = 0.032
|
$600
|
$19.20
|
|
WLWS
|
(0.4x0.5x0.4x0.1) = 0.008
|
$300
|
$2.40
|
|
WLSL
|
(0.4x0.5x0.1x0.5) = 0.01
|
-$100
|
-$1.00
|
|
WLSW
|
(0.4x0.5x0.1x0.4) = 0.008
|
$300
|
$2.40
|
|
WLSS
|
(0.4x0.5x0.1x0.1) = 0.002
|
$0
|
$0.00
|
|
WWLL
|
(0.4x0.4x0.5x0.5) = 0.04
|
$0
|
$0.00
|
|
WWLW
|
(0.4x0.4x0.5x0.4) = 0.032
|
$600
|
$19.20
|
|
WWLS
|
(0.4x0.4x0.5x0.1) = 0.008
|
$300
|
$2.40
|
|
WWWL
|
(0.4x0.4x0.4x0.5) = 0.032
|
$600
|
$19.20
|
|
WWWW
|
(0.4x0.4x0.4x0.4) = 0.0256
|
$1200
|
$30.72
|
|
WWWS
|
(0.4x0.4x0.4x0.1) = 0.0064
|
$900
|
$5.76
|
|
WWSL
|
(0.4x0.4x0.1x0.5) = 0.008
|
$300
|
$2.40
|
|
WWSW
|
(0.4x0.4x0.1x0.4) = 0.0064
|
$900
|
$5.76
|
|
WWSS
|
(0.4x0.4x0.1x0.1) = 0.0016
|
$600
|
$0.96
|
|
WSLL
|
(0.4x0.1x0.5x0.5) = 0.01
|
-$100
|
-$1.00
|
|
WSLW
|
(0.4x0.1x0.5x0.4) = 0.008
|
$300
|
$2.40
|
|
WSLS
|
(0.4x0.1x0.5x0.1) = 0.002
|
$0
|
$0.00
|
|
WSWL
|
(0.4x0.1x0.4x0.5) = 0.008
|
$300
|
$2.40
|
|
WSWW
|
(0.4x0.1x0.4x0.4) = 0.0064
|
$900
|
$5.76
|
|
WSWS
|
(0.4x0.1x0.4x0.1) = 0.0016
|
$600
|
$0.96
|
|
WSSL
|
(0.4x0.1x0.1x0.5) = 0.002
|
$0
|
$0.00
|
|
WSSW
|
(0.4x0.1x0.1x0.4) = 0.0016
|
$600
|
$0.96
|
|
WSSS
|
(0.4x0.1x0.1x0.1) = 0.0004
|
$300
|
$0.12
|
|
SL
|
(0.1x0.5) = 0.05
|
-$100
|
-$5.00
|
|
SWLL
|
(0.1x0.4x0.5x0.5) = 0.01
|
-$100
|
-$1.00
|
|
SWLW
|
(0.1x0.4x0.5x0.4) = 0.008
|
$300
|
$2.40
|
|
SWLS
|
(0.1x0.4x0.5x0.1) = 0.002
|
$0
|
$0.00
|
|
SWWL
|
(0.1x0.4x0.4x0.5) = 0.008
|
$300
|
$2.40
|
|
SWWW
|
(0.1x0.4x0.4x0.4) = 0.0064
|
$900
|
$5.76
|
|
SWWS
|
(0.1x0.4x0.4x0.1) = 0.0016
|
$600
|
$0.96
|
|
SWSL
|
(0.1x0.4x0.1x0.5) = 0.002
|
$0
|
$0.00
|
|
SWSW
|
(0.1x0.4x0.1x0.4) = 0.0016
|
$600
|
$0.96
|
|
SWSS
|
(0.1x0.4x0.1x0.1) = 0.0004
|
$300
|
$0.12
|
|
SSL
|
(0.1x0.1x0.5) = 0.005
|
-$100
|
-$0.50
|
|
SSWL
|
(0.1x0.1x0.4x0.5) = 0.002
|
$0
|
$0.00
|
|
SSWW
|
(0.1x0.1x0.4x0.4) = 0.0016
|
$600
|
$0.96
|
|
SSWS
|
(0.1x0.1x0.4x0.1) = 0.0004
|
$300
|
$0.12
|
|
SSSL
|
(0.1x0.1x0.1x0.5) = 0.0005
|
-$100
|
-$0.05
|
|
SSSW
|
(0.1x0.1x0.1x0.4) = 0.0004
|
$300
|
$0.12
|
|
SSSS
|
(0.1x0.1x0.1x0.1) = 0.0001
|
$0
|
$0.00
|
|
Total
|
1
|
|
$107.25
|
So you can see an
expected $17.85 (5.95%) loss is turned into a $107.25 (35.75%) profit by
using the bonus.
I don’t recommend betting 4 hands of $300 as the risks of not winning
become too large. As detailed in the step-by-step instructions you
should bet as much as you can until you have $1,000 in your account.
Then to reduce the risk of losing it all bet only $10 a hand until you
meet the gambling requirements. Appendix C shows some analysis on the
results of the “casino winners secrets” strategy under simulation.
Simulation Results
The simulation was done using SAS. SAS is a statistical analysis
software that is widely used in most companies that carry out data
analysis. It is also available in most universities. It is possibly not
the most efficient computer language for performing this type of
simulation however I use SAS in my work so it was the only language I
know well and was available to me.
Attached below is the actual program I used to produce the results. The
logic in it is as follows:
Start with a pot of $300, a stake of $300 and a gambling requirement
remainder (rem) of $1200
Using random numbers allocate a win [Pr(W) = 0.4], a [loss Pr(L) = 0.5]
and a stand off [Pr(S) = 0.1] to your hand
If the result is a win: pot = pot + stake
If the result is a loss: pot = pot – stake
If the result is a stand-off: pot = pot
Regardless of the result: rem = rem – stake
If the number of wins is less than 2 then the stake is set as follows
If the gambling requirement is met but the pot is $200 or less continue
gambling the entire pot.
If the gambling requirement or the pot is less than $500 then bet the
minimum of the two
Otherwise the stake should be $500
If the number of wins is 2 or greater (which should put you over $1,000
in the account) then reduce the stake to just $10.
Repeat steps 2 to 4 until rem = $0 or pot = $0.
If rem = $0 or pot = $0 then calculate the profit
If pot = $0 then profit = -$100
Otherwise profit = pot - $300
Repeat steps 1 – 6 for 24 casinos and calculate a total profit across
all 24
Repeat steps 1 – 7 for 2,000 simulations and determine average and
variance statistics.
The total profit across 24 casinos after 2,000 simulations produced the
following results.
Mean
$1,689.96 Std Deviation
1,661
Median
$1,800.00 Variance
2,759,128
Range
$13,940.00
Percentile
Estimate
100% Max $11,540
99%
$6,855
95%
$4,310
90%
$3,430
75% Q3
$2,630
50% Median $1,800
25% Q1
$930
10%
$10
5%
-$1,400
1%
-$2,400
0% Min
-$2,400
Extreme Observations
----Lowest----
----Highest----
Value
Obs Value
Obs
-$2,400
1993
$8,340
1823
-$2,400 1971
$8,680
734
-$2,400 1925
$9,460
1489
-$2,400 1915 $10,380 1341
-$2,400 1914 $11,540 856
What these results show is that
the most you can lose by going to all 24 casinos is $2,400 (ie. A loss
of $100 per casino). The most that was won was $11,540 and the average
winnings were $1,689.96.
The 10% percentile = $10.
This indicates you are likely to make $10 or more, 90% of the time.
The 25% percentile = $930 indicates that you are likely to make $930 or more
75% of the time. The 99% percentile = $6,855 indicates you are likely
to make $6,855 or more 1% of the time.
Remember that this is a fairly
conservative estimate. The probabilities I’ve applied to wins, losses
and stand-offs are only indicative but I have erred on the side of caution
so a simulation with more accurate probabilities could produce more
favorable results. I’ve provided the program so that you can test the
strategy and adjust it yourself.
The more casinos you try the more
likely you are to make a profit, however it also can result in a greater
loss. If you only want to go to 10 casinos then obviously the most you
can lose is $1,000. However the average return will be less.
I recommend forming a syndicate of
close friends to reduce the chance of sustaining a loss and increasing your
expected return.
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