The Logic of the Losing Shoe
For those in the casino industry, especially for us surveillance folks, the words ‘losing shoe’ are all too familiar.
A losing shoe is a period of play, normally lasting the length of one shoe of cards (which may be from one to as many decks as the shoe can hold!), which registers a substantial loss.
Ever wondered how that loss limit was set?
In considering this question, it is useful to once again refer to our central limit theorem. Here is the graph again.
(Source: http://schoolswikipedia.org/)
The central limit theorem proposes that up to 99.9% of all occurrences happen between 3 to 1 and 1 to 3 σs from the average or mean. σ extends into the positive (meaning 1 to 3 σ) and negative (meaning 1 to 3 σ).
In order to derive any sort of boundary in a casino game, one has to calculate the following:
 Probability Of Events Occurring (for more complex calculations, refer to: https://excelpunks.com/concepts8combinatorialanalysiscountingpossibleoutcomesandcreatingyourowncasinogame/)
 Mean Or Average Probability Of An Event Occurring (this is to determine the average expected result)
 Standard Deviation Of The Event Occurring (this is to determine σ)
Probability Of Events Occurring
The probability of events refers to the mathematical probability of an event. This can be calculated by determining the possible outcomes of what you are trying to measure and then dividing that by the total possible outcomes for an event.
A simple example would involve a single deck of cards. The probability of drawing an Ace of Spades is 1/52, since there is only one Ace of Spades in the deck. The probability of drawing an Ace of any suit is 4/52, since there are four Aces in the deck.
Mean Or Average Probability Of An Event Occurring
For this, we turn to binomial distribution. Binomial distribution calculates the means and standard deviations of occurrences that have one of two outcomes.
Example: Baccarat
Baccarat is a good example, where the outcome is either a Player or Banker. For simplicity, let’s disregard coups ending in a tie for now.
Formula for the mean (binomial distribution):
N x P = Number of trials of an event x probability of the event occurring
We generally know that the Banker has a probability of 0.458597 while the Player has that of 0.446247, with Ties making up the difference.
For a shoe where the player wagered exclusively on Banker for 40 coups, the mean win for him would be:
40 x 0.458597 = 18.34388 (this means that he would be expected to win 18.34388 coups)
Formula for the standard deviation (binomial distribution):
SQUARE ROOT(N x P x Q) = SQUARE ROOT (Number of trials of an event x probability of an event occurring x probability of an event NOT occurring)
The standard deviation for that kind of play as described above would be:
SQUARE ROOT(40 x 0.458597 x (10.458597)) = SQUARE ROOT(9.931431) =3.151417
*the greater the difference between P and Q, the smaller the standard deviation!
Standard Deviation Of The Event Occurring
Here is a table calculating the number of coups a player wagering exclusively on Banker would win from a 40 coup shoe in terms of σ.

Total Coups 
Mean 
σ 
3 σ 
2 σ 
1 σ 
1 σ 
2 σ 
3 σ 
40 
18.3438 
3.1514 
18.3438 – (3 x 3.1514) = 8.8896 coups 
18.3438 – (2 x 3.1514) = 12.041 coups 
18.3438 – (1 x 3.1514) = 15.1924 coups 
18.3438 + (1 x 3.1514) = 21.4952 coups 
18.3438 + (2 x 3.1514) = 24.6466 coups 
18.3438 + (3 x 3.1514) = 27.798 coups 

Now, to determine the expected result in dollars, we will multiply the player’s wager (assuming he wagered on ALL coups) by the coups he is expected to win or lose.
Determining Whether You Have a Losing Shoe
I have included 2 values, assuming average wagers of $1 and $5. You can add zeros to the backs of the average wagers as you please!

Percentage of Players Winning at that level 
2.50% 
13% 
34% 
50% 
34% 
13% 
2.50% 
Average Wager 
Standard Deviations 
3 σ 
2 σ 
1 σ 
Mean 
1 σ 
2 σ 
3 σ 
$1 
Winning Coups 
8.8896 
12.041 
15.1924 
18.3438 
21.4952 
24.6466 
27.798 
Losing Coups 
31.1104 
27.959 
24.8076 
21.6562 
18.5048 
15.3534 
12.202 
Expected Result 
($22.67) 
($16.52) 
($10.37) 
($4.23) 
$1.92 
$8.06 
$14.21 

Percentage of Players Winning at that level 
2.50% 
13% 
34% 
50% 
34% 
13% 
2.50% 
Average Wager 
Standard Deviations 
3 σ 
2 σ 
1 σ 
Mean 
1 σ 
2 σ 
3 σ 
$5 
Winning Coups 
8.8896 
12.041 
15.1924 
18.3438 
21.4952 
24.6466 
27.798 
Losing Coups 
31.1104 
27.959 
24.8076 
21.6562 
18.5048 
15.3534 
12.202 
Expected Result 
($113.33) 
($82.60) 
($51.87) 
($21.15) 
$9.58 
$40.30 
$71.03 
You can see that on average, players are expected to lose to the house. But when they start to win, the losing shoe is called – but at what point?
Notice the percentage levels above each σ. This means that for 2 σ, 13% of players would achieve a result of $16.52, wagering at $1 a coup. At the mean, 50% of players would achieve a loss of $4.23, wagering at $1 a coup.
To put that in perspective, a player wagering on Banker for 40 coups at $5,000 a coup would be expected to lose $21,150. If that player starts to win more than $9,580, you might want to watch him more closely. A win of $40,300 would be unlikely and a win of $71,030, even more so. You would definitely want to call a losing shoe at that point!
The ZTable
Now, the 3 to 3 σ measurement didn’t just pop out of nowhere. These measurements are derived from the Ztable.
Notice that the Ztable shows 0.0 to 3.0 down by the left and 0.00 to 0.09 on the top. This is the measurement in terms of σ, from 0.00 to 3.09.
You can determine the probability of σ by following the row on the left where the first 2 digits of your σ result appear to where it meets the column titled by the 3rd digit of your σ.
Examples:
0.16 σ would be 0.5636
1.82 σ would be 0.9656
The table is true for positive and negative σs, meaning that reading for 1.56 σ would be the same as 1.56 σ, which would be 0.9406.
You can set your own σ levels depending on your appetite for risk.
Good luck!
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