Predict Your Casino’s Future Earnings from Past Data – Linear Regression

Predict Your Casino’s Future Earnings from Past Data – Linear Regression Linear regression is a statistical tool for projecting future outcomes based on previous historical data.  A good example of this would be using the average wager to win/loss data from patrons to project how much a casino would be making for a particular period. Here’s the formula: Outcome = (Variable Multiplier x Variable) + Constant Or Y = bX + a This means that in order to find out the projected win/loss for a certain period, we would have to use a combination of a Constant Factor in addition to a Variable that is affected by a multiplier. We can find Y and X by getting the means of both data (the mean of the win/loss and average wager). We find b by using the following formula: b = r x (standard deviation of Y / standard deviation of X) r = sum of (individual X values – mean

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Concepts 9: Creating A Casino Game

Creating A Casino Game We’ve discussed the process for the creation of a casino game, namely: Calculating possibilities using combinatorial analysis Calculating the expectation Setting the odds Let’s use a simple example – something I saw in a Chinese drama. The game involves using 5 coins and tumbling them in an opaque container.  The container is then placed base up and players are then invited to make their wagers based on the permutations of the coins. Here are the calculations based on some of the possible wager types I can think of: Combinations Permutations Probability Expectation Break-Even Odds Total (5 + 2 -1!) / (5! X 2-1!) = 6 25 = 32 – – – 5 Tails 2! / 2! = 1 5! / 5! = 1 1/32 = 0.03125 0.03125 – (1-0.03125) = -0.9375 0.96875/0.03125 = 31 5 Heads 2! / 2! = 1 5! / 5! = 1 1/32 = 0.03125 0.03125 – (1-0.03125) = -0.9375 0.96875/0.03125 =

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Combinatorial Analysis – Counting Possible Outcomes AND creating YOUR OWN Casino Game!

We have seen how knowing the expectation enables us to set the odds of games.  If you recall, the formula for calculating the expectation for any wager is : (Probability of winning x the amount a wager would win) – (Probability of losing x the amount a wager would lose) So, if you wanted to create your own casino game, you would have to go through the following process: Calculating Probabilities of outcomes Calculating the Expectation Setting the Odds From the previous posts, you would have already figured out 2 and 3. How do we then get the probabilities of games?  This is perhaps the most fundamental aspect of casino games.  To do this, we would need to learn about combinatorial analysis. Combinatorial analysis is about counting possibilities.  This is essential when trying to determine the probabilities of events, such as a pair in Baccarat or a full house in poker.  Knowing how to calculate these probabilities then allows you

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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

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Check for a Biased Wheel, or a Dealer’s biased spin with Chi Square Tests

Chi2 The Chi2 distribution tests for the difference between the observed and the expected in terms of frequencies.  We can apply this to a simple example: In double-zero roulette, we have 38 numbers and the expected probability of any one of these numbers appearing is 1/38 or 0.026316 or 2.6316%.  Now, assuming you track all the results that appear on your roulette tables, you’d be able to check for biased wheels or even if your dealers have developed the muscle memory to spin at a regular area of the wheel. As with all things probable, do note that nothing is impossible.  It may be unlikely, but never impossible.  Always correlate your findings with footage from surveillance. Number Probability 0-0 0.026316 0 0.026316 1 0.026316 2 0.026316 3 0.026316 4 0.026316 5 0.026316 6 0.026316 7 0.026316 8 0.026316 9 0.026316 10 0.026316 11 0.026316 12 0.026316 13 0.026316 14 0.026316 15 0.026316 16 0.026316 17 0.026316 18 0.026316 19 0.026316

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Is Your New Player Suspicious? Scope him out using the Binomial Mean and Standard Deviation

Analyzing Gaming Results using the Binomial Mean and Standard Deviation In the previous post, we talked about the mean and standard deviation when analyzing the results of games. We discussed the Central Limit Theorem and how the majority of results of games would fall within the -1σ to 1σ region of a bell-curve. Here it is again, to jog your memory. (Source: http://schools-wikipedia.org/) Getting the mean and standard deviation of data is simple enough if you have a set of data – like when you are analyzing historical records of a current player. How about if you only have probabilities such as when you are analyzing casino games and assessing a completely new player? Mean Let’s use a coin-toss example. Each side of the coin has a 50% chance of appearing or a 0.5 probability.  If we tossed that coin 50 times, how many times are we expecting a heads or tails?  The calculations are as follows: Mean = number

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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. (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

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Maximise the Profitability of Your Casino by understanding Probability and Expectation in Gaming

Probability The basis of all games of chance is the concept of probability. What is probability?  Probability refers to the chance that an event will occur.  Probability is a value that ranges from 1, for an event that will definitely occur, to 0, for an event that will never occur. We get the probability of an event occurring by dividing the number of possible ways an event can occur by the total possible number of outcomes for all events.  Let’s illustrate: For a 6-sided die, we know that the total number of sides and numbers is 6.  So, for any single roll of the die, we could have any one of 6 possible outcomes.  6 is then the total possible number of outcomes for all events. Thus, the chance of rolling any number from 1 – 6 on a roll of the die is 1/6, assuming that the die isn’t loaded (we hope!). A variation of this, is the chance

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Find out if your latest marketing programme worked (or not) with the T-Test!

Do you need to compare results of an experiment or a marketing plan?  Have you just changed the settings of your slot machines and aren’t sure whether the change resulted in a real difference in earnings?  Have you just completed an intensive player promotion programme and need to find out whether it was effective? Statistics has a simple way of calculating if the revenue before and after resulted in a real change in performance and also the probability that the change was a DIRECT result of the changes you made!  This method is known as a T-Test. A T-Test compares the averages of 2 sets of values and assesses how different the sets of data are.  A companion to this is the R² measure of effect size, which calculates how much in terms of percentages that the results of your T-Test was affected directly by the change you implemented. Here’s an example: Casino A recently changed the settings of its

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