Tag Archives: analysis

Roulette Analysis

Requirements:
•Windows 7 and above
•.Net 4.5.2 compatibility
•SQL compatibility
Pricing: Let’s talk about it!
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You are a casino owner, gaming manager or surveillance manager.  You have a feeling that something is going down at the roulette tables, but you can’t put your finger on it.  Looking for a method to calculate the probabilities?
Excelpunks puts the statistics into this equation with Roulette Analysis!
•Includes analyses for Single and Double Zero roulette
•Probability alerts immediately inform you of improbable results.
•Round-By-Round Analysis and Entire Game Analysis:
  • Occurrence of Specific numbers
  • Win/Loss analysis on all wager types
  • Win/Loss analysis on Specific Numbers
  • Win/Loss analysis  of the game as a whole
•Custom Analysis •Win/Loss analysis of any combination of any number of games
Let robust statistical principles do the math for you.  Think that your player is winning more than he should?  Let Roulette Analysis prove it mathematically!
Get in touch with us at excelpunks@gmail.com for details!

Casino Surveillance Reporting Suite with Integrated Baccarat Analysis

Requirements:

  • Windows 7 and above
  • .NET Framework
  • SQL server compatibility

Contact us at excelpunks@gmail.com for details!

Demonstration Video:

Features:

  1. User-friendly interface with drag and drop functionality
  2. Comprehensive details presented in an easily navigable format.
  3. Compatibility with Excel – retrieve and upload large amounts of information using Excel into and out of the system.
  4. Customizable user access levels define how much access each user has to report and access data.
  5. Integrated Incident Reporting platform with Baccarat Analysis for comprehensive reporting on losing shoes.
  6. Automatic report data generates upon loading – presenting the number of reports created by respective users and categories.
  7. Print reports individually or by batch
Password Protected Log-ins
  • Password protection for all log-ins.
  • Reports are uniquely identified by the user’s log-in name.
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Incident Reporting

  • Standardized reporting made easy with drop-down lists for incident categories, locations, staff and subject profiles.
  • Dynamic report list generation allows you to scroll through all selected reports
Incident Report

Subject and Staff Profiles

  • Dynamic profile list generation allows you to scroll through all selected profiles
  • Attach any kind of file to the profile from images to movies and access them directly from the subject profile.
  • Easy access to all related Incident Reports and Daily Logs.

Subject Profile

Baccarat Analysis – Game Analysis

  • Examine a host of factors like wager ratios, game wins and shoe biases -all integrated with your Incident Report.

Baccarat Tracking Sheet

Baccarat Analysis – Player Analysis

  • Analyze all the games ever reviewed for specific players
  • Find out the level of probability for suspicious behavior for all reviewed games.

Player Analysis

Custom Report Generation

  • Custom report generation allows for dynamic, customized report lists for display.

Custom Data

Service

  • Customization is available on request – we understand how unique your operations are and we will tailor the system to suit your specific needs.

Should you be in another industry, but would like a customized reporting system of your own, do let us know!

The Logic Of The Losing Shoe

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.

bellcurve

(Source: http://schools-wikipedia.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:

  1. Probability Of Events Occurring (for more complex calculations, refer to: https://excelpunks.com/concepts-8-combinatorial-analysis-counting-possible-outcomes-and-creating-your-own-casino-game/)
  2. Mean Or Average Probability Of An Event Occurring (this is to determine the average expected result)
  3. 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 (1-0.458597)) = 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 Z-Table

Now, the -3 to 3 σ measurement didn’t just pop out of nowhere.  These measurements are derived from the Z-table.

Notice that the Z-table 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.

Z-table

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!

Compare How Different Areas of Your Casino are doing using an Independent T-Test!

What if you wanted to compare different areas with different numbers of machines on your gaming floor? What if you wanted to know if a certain area was receiving more profit than another?

In a previous post, we looked at how we can measure the effects of a change in machine settings on the earnings of a group of slot machines, using a dependent T-Test.

For the post, see here: https://excelpunks.com/comparisons-using-t-tests/

If you recall, a dependent T-test compares the differences in means of data with identical sample sizes before and after treatment – our data being the earnings of a slot machine and the treatment being a change in settings.

An independent T-test allows us to compare data sets with different numbers of samples to determine if they are statistically different.

Here’s an example:

A casino has 2 slot zones, Zone 1, with 5 machines and Zone 2, with 10 machines.  We want to find out if one area is earning more than another.

Here’s the data.

Zone 1 Machines Earnings Zone 2 Machines Earnings
1 $16 1 $8
2 $12 2 $10
3 $11 3 $15
4 $20 4 $15
5 $15 5 $14
6 $15
7 $12
8 $8
9 $13
10 $13

Step 1:

Calculate the mean earnings of each zone.

Zone 1 mean = (16 + 12 + 11 + 20 + 15) / 5 = $14.80

Zone 2 mean = (8 + 10 + 15 + 15 + 14 + 15 + 12 + 8 + 13 + 13) / 10 = $12.40

Step 2:

Subtract each machine’s individual earnings from their respective zone means and square the result.

Zone 1 Machines Earnings Earnings – Zone 1 Mean Square Zone 2 Machines Earnings Earnings – Zone 2 Mean Square
1 $16 16-14.80 = 1.2 $1.44 1 $8 8-12.40 = -4.4 $19.36
2 $12 12-14.80 = -2.8 $7.84 2 $10 10-12.40 = -2.4 $5.76
3 $11 11-14.80 = -3.8 $14.44 3 $15 15-12.40 = 2.6 $6.76
4 $20 20-14.80 = 5.2 $27.04 4 $15 15-12.40 = 2.6 $6.76
5 $15 15-14.80 = 0.2 $0.04 5 $14 14-12.40 = 1.6 $2.56
6 $15 15-12.40 = 2.6 $6.76
7 $12 12-12.40 = -0.4 $0.16
8 $8 8-12.40 =  -4.4 $19.36
9 $13 13-12.40 = 0.6 $0.36
10 $13 13-12.40 = 0.6 $0.36

Step 3:

Now add the results together and divide them by the total number of samples -2.

Zone 1 = 1.44 + 7.84 + 14.44 + 27.04 + 0.04 = 50.80

Zone 2 = 19.36 + 5.76 + 6.76 + 6.76 + 2.56 + 6.76 + 0.16 + 19.36 + 0.36 + 0.36 = 68.20

50.80 + 68.20 = 119

119 / 5+10-2 = 119/13 = 9.15 (this is known as the Pooled Variance)

Step 4:

Divide the pooled variance by the respective sample sizes of the 2 zones, add the result and finally square root that.

(9.15 / 5) + (9.15/10) = 1.83 + 0.92 = 2.75

Square Root 2.75 = 1.657

Step 5:

Now, we take the difference of the means for Zone 1 and 2 and divide it by 1.657.  This is known as your T-statistic.

T = 14.80 – 12.40 / 1.657 = 2.40 / 1.657 = 1.448267583 or 1.448

Step 6 (Finally):

Compare the T-statistic with the T-table.  Looking at this table, we get our degrees of freedom, or df, by subtracting 2 from 5+10 = 13.  Now, we will find our T-statistic on this table.

Ind T

We see that our T-statistic of 1.448 is between 90% – 95% on a one-tailed test.  This means that we are between 90% – 95% sure that from this sample data, Zone 1’s earnings are higher than Zone 2’s earnings.

Perhaps you might want to place more machines in Zone 1 then?

Here’s the formula for an independent T-test:

T = mean1 – mean2 / sqrt((pooled variance / n1) + (pooled variance / n2))

Here’s a spreadsheet that allows you to compare data using the Independent T-test.  Just paste your data in the columns A and B and click ‘Compute’!

https://drive.google.com/file/d/0B1pEq2dN7H9Aa3RTdng3b1d1M00/view?usp=sharing

Roulette – An Analysis

Roulette – An Analysis

Roulette is played by spinning a ball around a wheel.  The wheel contains 37 to 38 numbered pockets from 0 to 36 for a 37-numbered wheel and 00 and 0 to 36 for a 38-numbered wheel.

The winning number is decided by which numbered pocket the ball comes to rest on.  Payment is then made based on where a player has wagered on a layout displaying all the numbers on the wheel.

This table assesses the wagers for Single-Zero (European) Roulette:

Double-0

European Roulette (Single Zero)
Wager Numbers covered Probability (Win) Probability (Loss) Expectation Break-even odds
Straight Up 1 0.02703 0.97297 -0.94595 36.00
Split 2 0.05405 0.94595 -0.89189 17.50
Street 3 0.08108 0.91892 -0.83784 11.33
Corner 4 0.10811 0.89189 -0.78378 8.25
6-Line 6 0.16216 0.83784 -0.67568 5.17
Dozen 12 0.32432 0.67568 -0.35135 2.08
Column 12 0.32432 0.67568 -0.35135 2.08
Small (1-18) 18 0.48649 0.51351 -0.02703 1.06
Big (19-36) 18 0.48649 0.51351 -0.02703 1.06
Red 18 0.48649 0.51351 -0.02703 1.06
Black 18 0.48649 0.51351 -0.02703 1.06
Even 18 0.48649 0.51351 -0.02703 1.06
Odd 18 0.48649 0.51351 -0.02703 1.06

This table assesses the wagers for Double-Zero (American) Roulette:

Single-0

American Roulette (Double Zero)
Wager Numbers covered Probability (Win) Probability (Loss) Expectation Break-even odds
Straight Up 1 0.02632 0.97368 -0.94737 37.00
Split 2 0.05263 0.94737 -0.89474 18.00
Street 3 0.07895 0.92105 -0.84211 11.67
Corner 4 0.10526 0.89474 -0.78947 8.50
6-Line 6 0.15789 0.84211 -0.68421 5.33
Dozen 12 0.31579 0.68421 -0.36842 2.17
Column 12 0.31579 0.68421 -0.36842 2.17
Small (1-18) 18 0.47368 0.52632 -0.05263 1.11
Big (19-36) 18 0.47368 0.52632 -0.05263 1.11
Red 18 0.47368 0.52632 -0.05263 1.11
Black 18 0.47368 0.52632 -0.05263 1.11
Even 18 0.47368 0.52632 -0.05263 1.11
Odd 18 0.47368 0.52632 -0.05263 1.11

Wager lists the specific wager type players can make on the layout.

Numbers covered indicates how many numbers the wager will be counted to be wagered on.  If the ball rests on any of the numbers covered by this wager, the wager is paid.

Probability (Win) indicates the probability that this wager will win. This is calculated by:

= Numbers covered/total numbers on the layout (37 for Single-Zero and 38 for Double-Zero)

Example:

For a Corner (Single-Zero) wager, 4/37 = 0.10811

Probability (Loss) indicates the probability that the wager will lose.  The calculation for this is 1-Probability (Win).

Example:

For a Street (Double-Zero) wager, 3/38 = 1-0.07895 = 0.92105

Expectation is the Probability (Win) – Probability (Loss).

Example:

For a Street (Double-Zero) wager, 0.07895 – 0.92105= -0.84211

Break-even odds indicates the odds at which the expectation for the house is 0.  The house should ALWAYS pay BELOW these odds to maintain a positive expectation.

Sic-Bo and Dice – An Analysis

Sic-Bo is really popular in Asia.  This game involves 3 dice in a tumbler or variant which is tumbled, with varying payments for specific combinations of the dice.

There is an amazing way to calculate how many possible outcomes for a particular score from 3 six-sided dice there can be.  This involves the use of polynomial multiplication.

Here’s a simple way to look at this…

You have 3 dice with numbers from 1 to 6 on each of them.

Die 1 1 2 3 4 5 6
Die 2 1 2 3 4 5 6
Die 3 1 2 3 4 5 6

Now, add each number from Die 1 to the first number of Die 2 and then the second and so on like this.

Die 1
Die 2 1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12

Now, add each of these 36 results to the first number of Die 3 and then the second and so on like this.

Die 3
Die 1 and 2 results 1 2 3 4 5 6
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
7 8 9 10 11 12 13
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
7 8 9 10 11 12 13
8 9 10 11 12 13 14
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
7 8 9 10 11 12 13
8 9 10 11 12 13 14
9 10 11 12 13 14 15
5 6 7 8 9 10 11
6 7 8 9 10 11 12
7 8 9 10 11 12 13
8 9 10 11 12 13 14
9 10 11 12 13 14 15
10 11 12 13 14 15 16
6 7 8 9 10 11 12
7 8 9 10 11 12 13
8 9 10 11 12 13 14
9 10 11 12 13 14 15
10 11 12 13 14 15 16
11 12 13 14 15 16 17
7 8 9 10 11 12 13
8 9 10 11 12 13 14
9 10 11 12 13 14 15
10 11 12 13 14 15 16
11 12 13 14 15 16 17
12 13 14 15 16 17 18

Now, by counting the number of instances that each number appears in this table, we would have the number of possible outcomes for each score.  This works on any number of dice.

Points Number of possible outcomes Probability (Win) Probability (Loss) Expectation Break-Even Odds
3 1 0.0046 0.9954 -0.9907 215.00
4 3 0.0139 0.9861 -0.9722 71.00
5 6 0.0278 0.9722 -0.9444 35.00
6 10 0.0463 0.9537 -0.9074 20.60
7 15 0.0694 0.9306 -0.8611 13.40
8 21 0.0972 0.9028 -0.8056 9.29
9 25 0.1157 0.8843 -0.7685 7.64
10 27 0.1250 0.8750 -0.7500 7.00
11 27 0.1250 0.8750 -0.7500 7.00
12 25 0.1157 0.8843 -0.7685 7.64
13 21 0.0972 0.9028 -0.8056 9.29
14 15 0.0694 0.9306 -0.8611 13.40
15 10 0.0463 0.9537 -0.9074 20.60
16 6 0.0278 0.9722 -0.9444 35.00
17 3 0.0139 0.9861 -0.9722 71.00
18 1 0.0046 0.9954 -0.9907 215.00

The following table lists the most common wager types and their respective details.

Wager Number of possible outcomes Probability (Win) Probability (Loss) Expectation Break-Even Odds
Specific Triple 6 combinations and 1 possible outcome each 0.0046 0.9954 -0.9907 215
Any Triple 6 0.0278 0.9722 -0.9444 35.00
Specific Doubles 16 (15 + 1(the specific triple of that number)) 0.0741 0.9259 -0.8519 12.50
Big/Small 105 (108 possible outcomes – 111,222,333 (for small) ,444,555 and 666 (for big)) 0.4861 0.5139 -0.0278 1.06
Even/Odd 105 (108 possible outcomes – 111,333,555 (for odd) ,222,444 and 666 (for even)) 0.4861 0.5139 -0.0278 1.06
2 Dice and Single Die Combination 30 combinations and 3 possible outcomes for each 0.0138 0.9861 -0.9722 71
3 Single Die Combination 20 combinations and 6 possible outcomes for each 0.0277 0.9722 -0.9444 35

Points refers to the total value from a 3-dice roll, from 3 to 18.

The number of possible outcomes is derived from tabulating all possible outcomes from a 3-dice roll. The total number of possible outcomes is 216 or 6 x 6 x 6 (this is the formula for a permutation where repetition is allowed).

Probability (Win) refers to the possible outcomes for each number divided by the total possible outcomes from a 3-dice roll.

Example:

For point 5, 6/216 = 0.0278

For point 10, 27/216 = 0.1250

Probability (Loss) is 1-Probability (Win).

Example:

For point 5, 1-0.0278=0.9722

For point 9, 1-0.1157=0.8843

Expectation is Probability (Win) – Probability (Loss).

Example:

For point 9, 0.1157 – 0.8843 = -0.7685

In the table the expectation has a ‘-‘ in front as this is in the perspective of the player having a negative expectation; which is positive for the house.

Break-even odds refers to the payment odds at which expectation is 0.  This means that the house should ALWAYS pay BELOW these odds in order to maintain a positive expectation for the house.

The is derived from the Probability (Loss)/ Probability (Win),

Example:

For point 3,  0.9954/0.0046 = 215

For point 6, 0.9537/0.0463 = 20.6

After computing all the possible outcomes of a 3-dice roll, we can see that the possible outcomes and probability follow the central limit theorem nicely.  The expectation dovetails the probability, being the result of 1-probability.

Number of possible outcomes for each point value from 3 to 18

Sic-Bo1

Probability for each point value from 3 to 18

Sic-Bo2

Expectation for each point value from 3 to 18

Sic-Bo3

The following table shows a similar tabulation for 3-Single Dice combinations and Double and Single Dice combinations.  Again, the total possible number of outcomes is 216, with the number of categories being 56:

S/No Combination Possible Outcomes Probability (Win) Probability (Loss) Expectation Break-even odds
1 666 1 0.0046 0.9954 -0.9907 215
2 333 1 0.0046 0.9954 -0.9907 215
3 111 1 0.0046 0.9954 -0.9907 215
4 444 1 0.0046 0.9954 -0.9907 215
5 222 1 0.0046 0.9954 -0.9907 215
6 555 1 0.0046 0.9954 -0.9907 215
7 244 3 0.0139 0.9861 -0.9722 71
8 334 3 0.0139 0.9861 -0.9722 71
9 144 3 0.0139 0.9861 -0.9722 71
10 335 3 0.0139 0.9861 -0.9722 71
11 166 3 0.0139 0.9861 -0.9722 71
12 336 3 0.0139 0.9861 -0.9722 71
13 223 3 0.0139 0.9861 -0.9722 71
14 344 3 0.0139 0.9861 -0.9722 71
15 225 3 0.0139 0.9861 -0.9722 71
16 355 3 0.0139 0.9861 -0.9722 71
17 233 3 0.0139 0.9861 -0.9722 71
18 366 3 0.0139 0.9861 -0.9722 71
19 114 3 0.0139 0.9861 -0.9722 71
20 116 3 0.0139 0.9861 -0.9722 71
21 266 3 0.0139 0.9861 -0.9722 71
22 445 3 0.0139 0.9861 -0.9722 71
23 155 3 0.0139 0.9861 -0.9722 71
24 446 3 0.0139 0.9861 -0.9722 71
25 224 3 0.0139 0.9861 -0.9722 71
26 455 3 0.0139 0.9861 -0.9722 71
27 113 3 0.0139 0.9861 -0.9722 71
28 466 3 0.0139 0.9861 -0.9722 71
29 115 3 0.0139 0.9861 -0.9722 71
30 122 3 0.0139 0.9861 -0.9722 71
31 226 3 0.0139 0.9861 -0.9722 71
32 556 3 0.0139 0.9861 -0.9722 71
33 112 3 0.0139 0.9861 -0.9722 71
34 566 3 0.0139 0.9861 -0.9722 71
35 255 3 0.0139 0.9861 -0.9722 71
36 133 3 0.0139 0.9861 -0.9722 71
37 235 6 0.0278 0.9722 -0.9444 35
38 236 6 0.0278 0.9722 -0.9444 35
39 135 6 0.0278 0.9722 -0.9444 35
40 125 6 0.0278 0.9722 -0.9444 35
41 124 6 0.0278 0.9722 -0.9444 35
42 245 6 0.0278 0.9722 -0.9444 35
43 346 6 0.0278 0.9722 -0.9444 35
44 456 6 0.0278 0.9722 -0.9444 35
45 356 6 0.0278 0.9722 -0.9444 35
46 246 6 0.0278 0.9722 -0.9444 35
47 146 6 0.0278 0.9722 -0.9444 35
48 156 6 0.0278 0.9722 -0.9444 35
49 345 6 0.0278 0.9722 -0.9444 35
50 256 6 0.0278 0.9722 -0.9444 35
51 234 6 0.0278 0.9722 -0.9444 35
52 126 6 0.0278 0.9722 -0.9444 35
53 145 6 0.0278 0.9722 -0.9444 35
54 123 6 0.0278 0.9722 -0.9444 35
55 136 6 0.0278 0.9722 -0.9444 35
56 134 6 0.0278 0.9722 -0.9444 35

So, how do you calculate the combinations and permutations mathematically?

Here’s a great method created by Dr. James Tanton, known as labelling – it’s effective and easy to learn!  (http://gdaymath.com/courses/permutations-and-combinations/)

Combinations

For combinations, we use 6 factorial or 6! as our numerator (due to the die being 6-sided). Our denominator would be a multiplication of the numbers involved in the combination and those not, with the total being 6 as well. Here’re some examples:

Specific Triples: 6! / (1! X 5!) = 720 / (1 x 120) = 6 combinations. (Notice that 6! / (1! X 5!) has the numerator equalling 6 and the denominator also equalling 6 – that’s how you check the maths!)

Remember that there are 6 numbers, so we have 6/6 = 1 combination for each number. This would apply in the next example too.

Specific Doubles: 6! / (1! X 1! X 4!) = 720 / (1 x 1 x 24) = 30 combinations (Notice how the numerator and denominator also equal 6? Notice the denominator 1! X 1! X 4!, as 1 of the numbers would be a double, 1 of the numbers would be a single and the other 4 numbers would not be in the combination.)

Remember that for Specific Doubles there are 6 numbers, so each number would have 30 / 6 = 5 combinations for each number.

3 Single Die Combination: 6!/(3! X 3!) = 720 / (6 x 6) = 720 / 36 = 20 combinations. (We have 3! as 3 of the numbers of the 6 are singles included in the combination, with the other 3 numbers out of the combination.)

Total Combinations: (3 + 6 – 1)! / (3! x (6 – 1)! = 8!/(3! x 5!) = 40,320 / 720 = 56 total combinations

Permutations

For permutations, we use 3 factorial or 3! as our numerator (due to there being 3 dice in the game).

Specific Triples: 3!/3! = 6/6 = 1 permutation per combination (our denominator is 3! as all 3 numbers would have to be the same in order to make a triple.)

Specific Doubles: 3!/2! X 1! = 6/2 = 3 permutations per combination (our denominator is 2! X 1! as 2 of the numbers must be the same to make a double, with the last being any other number.)

3 Single Die Combination: 3!/(1! X 1! X 1!) = 6 permutations per combination (our denominator is 1! X 1! X 1! as all 3 numbers have to be difference to make a 3 die combination.)

Total permutations: 6 x 6 x 6 = 216 total permutations

Appendix: Here is a table listing all the possible outcomes on Sic-Bo.

D1 D2 D3 Points Combination
1 1 1 3 111
2 1 1 4 112
3 1 1 5 113
4 1 1 6 114
5 1 1 7 115
6 1 1 8 116
1 2 1 4 112
2 2 1 5 122
3 2 1 6 123
4 2 1 7 124
5 2 1 8 125
6 2 1 9 126
1 3 1 5 113
2 3 1 6 123
3 3 1 7 133
4 3 1 8 134
5 3 1 9 135
6 3 1 10 136
1 4 1 6 114
2 4 1 7 124
3 4 1 8 134
4 4 1 9 144
5 4 1 10 145
6 4 1 11 146
1 5 1 7 115
2 5 1 8 125
3 5 1 9 135
4 5 1 10 145
5 5 1 11 155
6 5 1 12 156
1 6 1 8 116
2 6 1 9 126
3 6 1 10 136
4 6 1 11 146
5 6 1 12 156
6 6 1 13 166
1 1 2 4 112
2 1 2 5 122
3 1 2 6 123
4 1 2 7 124
5 1 2 8 125
6 1 2 9 126
1 2 2 5 122
2 2 2 6 222
3 2 2 7 223
4 2 2 8 224
5 2 2 9 225
6 2 2 10 226
1 3 2 6 123
2 3 2 7 223
3 3 2 8 233
4 3 2 9 234
5 3 2 10 235
6 3 2 11 236
1 4 2 7 124
2 4 2 8 224
3 4 2 9 234
4 4 2 10 244
5 4 2 11 245
6 4 2 12 246
1 5 2 8 125
2 5 2 9 225
3 5 2 10 235
4 5 2 11 245
5 5 2 12 255
6 5 2 13 256
1 6 2 9 126
2 6 2 10 226
3 6 2 11 236
4 6 2 12 246
5 6 2 13 256
6 6 2 14 266
1 1 3 5 113
2 1 3 6 123
3 1 3 7 133
4 1 3 8 134
5 1 3 9 135
6 1 3 10 136
1 2 3 6 123
2 2 3 7 223
3 2 3 8 233
4 2 3 9 234
5 2 3 10 235
6 2 3 11 236
1 3 3 7 133
2 3 3 8 233
3 3 3 9 333
4 3 3 10 334
5 3 3 11 335
6 3 3 12 336
1 4 3 8 134
2 4 3 9 234
3 4 3 10 334
4 4 3 11 344
5 4 3 12 345
6 4 3 13 346
1 5 3 9 135
2 5 3 10 235
3 5 3 11 335
4 5 3 12 345
5 5 3 13 355
6 5 3 14 356
1 6 3 10 136
2 6 3 11 236
3 6 3 12 336
4 6 3 13 346
5 6 3 14 356
6 6 3 15 366
1 1 4 6 114
2 1 4 7 124
3 1 4 8 134
4 1 4 9 144
5 1 4 10 145
6 1 4 11 146
1 2 4 7 124
2 2 4 8 224
3 2 4 9 234
4 2 4 10 244
5 2 4 11 245
6 2 4 12 246
1 3 4 8 134
2 3 4 9 234
3 3 4 10 334
4 3 4 11 344
5 3 4 12 345
6 3 4 13 346
1 4 4 9 144
2 4 4 10 244
3 4 4 11 344
4 4 4 12 444
5 4 4 13 445
6 4 4 14 446
1 5 4 10 145
2 5 4 11 245
3 5 4 12 345
4 5 4 13 445
5 5 4 14 455
6 5 4 15 456
1 6 4 11 146
2 6 4 12 246
3 6 4 13 346
4 6 4 14 446
5 6 4 15 456
6 6 4 16 466
1 1 5 7 115
2 1 5 8 125
3 1 5 9 135
4 1 5 10 145
5 1 5 11 155
6 1 5 12 156
1 2 5 8 125
2 2 5 9 225
3 2 5 10 235
4 2 5 11 245
5 2 5 12 255
6 2 5 13 256
1 3 5 9 135
2 3 5 10 235
3 3 5 11 335
4 3 5 12 345
5 3 5 13 355
6 3 5 14 356
1 4 5 10 145
2 4 5 11 245
3 4 5 12 345
4 4 5 13 445
5 4 5 14 455
6 4 5 15 456
1 5 5 11 155
2 5 5 12 255
3 5 5 13 355
4 5 5 14 455
5 5 5 15 555
6 5 5 16 556
1 6 5 12 156
2 6 5 13 256
3 6 5 14 356
4 6 5 15 456
5 6 5 16 556
6 6 5 17 566
1 1 6 8 116
2 1 6 9 126
3 1 6 10 136
4 1 6 11 146
5 1 6 12 156
6 1 6 13 166
1 2 6 9 126
2 2 6 10 226
3 2 6 11 236
4 2 6 12 246
5 2 6 13 256
6 2 6 14 266
1 3 6 10 136
2 3 6 11 236
3 3 6 12 336
4 3 6 13 346
5 3 6 14 356
6 3 6 15 366
1 4 6 11 146
2 4 6 12 246
3 4 6 13 346
4 4 6 14 446
5 4 6 15 456
6 4 6 16 466
1 5 6 12 156
2 5 6 13 256
3 5 6 14 356
4 5 6 15 456
5 5 6 16 556
6 5 6 17 566
1 6 6 13 166
2 6 6 14 266
3 6 6 15 366
4 6 6 16 466
5 6 6 17 566
6 6 6 18 666

Understand Your Casino: Mean and Standard Deviation in Gaming

Mean and Standard Deviation in Gaming

In the previous post, we talked about the long term expectation of games derived from the probabilities of outcomes in casino games.

How about measuring results in the short-term such as when the results of shoes or player activity appear inconsistent with the expected values?  Let’s face it, they usually do!

For this, it is useful to apply the principles of the central limit theorem, in particular, the concepts of mean and standard deviation.

But first…

Central Limit Theorem

The central limit theorem explains that the outcome of repeated experiments will follow a bell-shaped pattern.  In terms of casino games, we can translate this as meaning that the outcomes of play will follow a certain pattern of winnings and losses.

bellcurve

(Source: http://schools-wikipedia.org/)

This is a bell-curve, so named due to its shape.

The bell-curve’s centre is known as the mean or average (µ).

Notice that the bell-curve is divided into 6 parts from -3σ at the extreme left to 3σ at the extreme right.  The σ symbol means standard deviation.  Thus, the bell-curve is divided into units of 6 standard deviations.

The central limit theorem explains that for an ideally NORMAL situation, approximately 68.2% (34.1% on either side) of all outcomes will occur within the -1σ to 1σ region, with approximately 27.2% of all outcomes occurring between the -1σ and -2σ regions (13.6%) and the 1σ and 2σ regions (13.6% as well).  The remaining 5% thereabouts is divided into two and occur between the -2σ and -3σ and 2σ and 3σ regions.

From the bell-curve, assuming you believe it, we can assume that the majority of outcomes will occur within the -1σ to the 1σ region.

We can translate this into casino gaming, where we can infer that this bell-curve would also apply to the results of games.

So, how would we know?

Mean

The most important part to this question lies in determining the mean or average of the data.  Let’s take this player report as a sample:

Player Win/(Loss)
Player 1 $100
Player 2 ($500)
Player 3 ($300)
Player 4 $200
Player 5 ($100)

The mean of the win/loss for all 5 players is = $100 + (-$500) + (-$300) + $200 + (-$100) / 5 = -$600 / 5 = -$120

This means that on average, these players lost $120 each.  So, -$120 becomes our middle point in our bell-curve.

Standard Deviation

Now, assuming that an average $120 loss is also true for all players, we can now find the standard deviation.  The calculation is as follows:

Player Win/(Loss) Win/ (Loss) – Mean (Win/ (Loss) – Mean)2 Variance Standard Deviation = √Variance
Player 1 $100 $100 – ($120) = $220 $220 x $220 = $48,400 $48,400 + $144,400 + $32,400 + $102,400 + $400 / (5-1) =$328,000 / 4 = $82,000 Square root ($82,000) = $286.3564
Player 2 ($500) ($500) – ($120) = -$380 -$380 x -$380 = $144,400
Player 3 ($300) ($300) – ($120) = – $180 -$180 x -$180 = $32,400
Player 4 $200 $200 – ($120) = $320 -$320 x -$320 = $102,400
Player 5 ($100) ($100) – ($120) = $20 $20 x $20 = $400
Mean ($120)

With a standard deviation of $286.3564 or $286.36, we can apply the probabilities in the bell-curve to the results of games in the short-term.

Probability 2.3% 13.6% 34.1% 50% 34.1% 13.6% 2.3%
Standard Deviations -3σ -2σ -1σ Mean
Result -3 x $286.36 + (-$120) =-$979.08 -2 x $286.36 + (-$120) =-$692.72 -1 x $286.36 + (-$120)=-$406.36 -$120 1 x $286.36 + (-$120) = $166.36 2 x $286.36 + (-$120) = $452.72 3 x $286.36 + (-$120) = $739.08

What this now means, is that based on our data, the chance of a player losing $120 is 50% and most players will win between $166.36 or lose $406.36.  Wins or losses outside of this range are less likely and thus, more suspect.

This being the case, we actually now have Player 4 winning $200, which is more than the expected win of $166.36 – we should probably have a look at his play.

How we will do that will be discussed in a later post.