By Dale S. Yeazel

The Law of Large Numbers (Law of “Averages”)

Now, don’t be scared. You probably know as much about “The law of large numbers” as you need to. Either intellectually or intuitively, you know that the greater the number of trials, the closer the results will resemble what probability says they should be. And if you don’t, here is an example from the book.

The probability of throwing “x” number of heads in “n” number of throws, can be calculated using the “binomial probability distribution.” The binomial formula is given by:

P_x = n!/[x!(n-x)!]p^x(1-p)^n-x which I find easier to understand in this format:

I don’t know about you but I’ve taken all the shit I’m going to take from these formulas. I’m sure we can work out a problem using this formula. Remember:

P or p = the probability of the event.

n = the number of throws (events).

x = the number of heads.

Lets do the example of 6 heads in 10 throws. So, p=.5, n=10 and x=6

P_x becomes P₆ and means “the probability of six heads.” The only other problem I can see you having is with the last pair of parentheses: (1-p)^n-x which translates to (1-.5)^10-6 or .5⁴

P_x = n!/[x!(n-x)!]p^x(1-p)^n-x

P₆ = 10!/[6!(10-6)!]p⁶(1-.5)^10-6 =

3,628,800/17,280*.015625*.0625 =

210*.015625*.0625 = .2050781250000005

But now comes the mystery. Why doesn’t this answer match the one (.377) given in the book? The reason is: I computed the probability of throwing exactly six heads. To get the final answer I must add the chances of throwing exactly 7, 8, 9 and 10 heads in 10 tosses.

And no, I didn’t compute all those combinations out by hand. I found another online calculator that you will find very helpful.

http://faculty.vassar.edu/lowry/binomialX.html

You enter the number of trials (n) the number of desired results (they use “k”) and the probability of the result (p) that in the case of our coin toss is .5

But let me save you some aggravation and show you how to solve some problems that are common for the work we are doing. You do need to know about the “shift” key in the upper left corner. Unlike a typewriter, you press (but not hold) the shift key in order to access the function that is written on the case itself, just above the key.

Simplify 6!

[6] [Shift] [x!] [=] the equal sign in on the bottom right of the calculator. Your result should equal 720.

Simplify 4⁵

[4] [^] [5] [=] and the answer should be 1024.

5P2 or P_5,2

[5] [Shift] [nPr] [2] [=] and you should come up with 20.

5C2 or C_5,2

[5] [Shift] [nCr] [2] [=] and your answer should be 10.

What is the square root of 144?

[√] [144] [=] and you should get 12.

Central Limit Theorem

Theorem – 1. An idea that is demonstrably true or is assumed to be so. 2. Mathematics a. A proposition that is provable on the basis of explicit assumptions, b. A proven proposition.

Theorem – A statement that can be demonstrated to be true by accepted mathematical operations and arguments.

“Among the most powerful theorems in probability and statistics is the central limit theorem, the central limit theorem states that for a large number of independent trials (wagers), the distribution of the total, or percentage of the total, can be described by the normal curve.”

I have probably looked at more bell curves than Quasimodo but they made as much sense to me as an oscilloscope would to a caveman. I then began to understand the purpose of them, which is to provide a method of estimating fluctuation based on expectation of an event and the number of trials.

Imagine that the red ruler on the bottom can be slid left or right. We have slid it to the left so that we may center the bell curves on the HA for a bet on red in roulette (5.56). The first thing you must recognize about those three curves is this: the taller they are, the greater the number of trials (n). And second, the narrower they are, the closer the results are to the expected value of 5.56%.

“Since the normal curve has been well studied, assessing probabilities for any phenomenon that can be described with this curve is possible. In particular it can gauge how close, in terms of probability, the actual percentage casino win will be to the theoretical or expected win.”

Normal probability distribution

But I found an even better book on probability in general and bell curves in particular. It is “Probability Demystified” by Allan G. Bluman

“Many continuous variables can be represented by formulas and graphs or curves. These curves represent probability distributions. In order to find probabilities for values of a variable, the area under the curve between two given values is used.” (Don’t worry if you don’t understand this, you will see it in action soon).

“One of the most often used continuous probability distributions is called the normal probability distribution. Many variables are approximately normally distributed and can be represented by the normal distribution. It is important to realize that the normal distribution is a perfect theoretical mathematical curve but no real-life variable is perfectly normally distributed.”

I think it is important not to lose sight of what we are talking about here: variables. A variable might be anything from a casino’s hold percentage to how many people will go to the movies tonight. And these variables can be estimated by using the normal probability distribution.

“The real-life normally distributed variables can be described by the theoretical normal distribution. This is not so unusual when you think about it. Consider the wheel. The mathematically perfect circle can represent it. But no real-life wheel is perfectly round. The mathematics of the circle, then, is used to describe the wheel.”

In other words, it isn’t really surprising that theoretical formulas can represent real-life situations since these formulas have been well researched, even before the advent of the computer. And don’t forget about the law of large numbers. Only with a sufficiently large number of trials, does real life resemble normal probability distribution.

The normal distribution has the following properties:

1.) It is bell shaped.

2.) The mean, median and mode are at the center of the distribution.

3.) It is symmetric about the mean. (This means that it is a reflection of itself, if a mean was placed at the center).

4.) It is continuous; i.e., there are no gaps.

5.) It never touches the x-axis (the horizontal line at the bottom of the graph).

6.) The total area under the curve is 1 or 100%.

7.) About 0.68 or 68% of the area under the curve falls within one standard deviation on either side of the mean. (Recall that m is the symbol for the mean and s is the symbol for standard deviation.

About 1.00 or 100% of the area falls within three standard deviations of the mean. (Note: It is somewhat less than 100%, but for simplicity, 100% will be used here).

Based on the earlier requirements for a normal distribution, these are my observations from the above bell curve:

Here is a probability problem from “Probability Demystified” that illustrates how you can use the above bell curve to solve it:

EXAMPLE:

The mean commuting time between a person’s home and office is 24 minutes. The standard deviation is 2 minutes. Assume the variable is normally distributed. Find the probability that it takes a person between 24 and 28 minutes to get to work.

SOLUTION:

Draw the normal distribution and place the mean, 24, at the center. Then place the mean plus one standard deviation (26) to the right, the plus two standard deviations (28) to the right, the mean plus three standard deviation (30) to the right, the mean minus one standard deviation (22) to the left, the mean minus two standard deviations (20) to the left and the mean minus three standard deviations (18) to the left.

Using the shaded areas highlighting the area under the curve that is between 24 and 28 minutes, we can see that the probability of the commute taking that long is:

0.341 + 0.136 = 0.477 or 47.7%.

Volatility, Standard Deviation and Confidence Levels

We stand at the threshold of the very alter of divine gaming knowledge. I mean; this is the kind of stuff they discuss at those fancy shift boss meetings, I think. The ramification of the impact this material can have to your education and thus your career could send shockwaves that will be felt throughout the quadrant. And anyway, we have made it this far; it would be a shame to quit now.

Volatility

No, “volatility” isn’t a reference to your boss’s mood swing when your table is dumping. Volatility is the amount of deviation (or variation) in a game from what is considered to be expected results. This is what the bell curve was demonstrating, that the greater the number of trials, the less the amount of volatility (deviation) will occur. When you think about it, volatility is the reason people gamble. If the results of wagers always followed their expected results for the long run, games of chance would be very boring affairs.

Volatility is measured in terms of “standard deviation” and these two variables are the main ingredients in “volatility analysis.” “Volatility analysis” is the Sherlock Holmes like technique that enables managers to know if “something is rotten in the State of Denmark.”

In Mario Puzo’s novel, “Fools Die” a casino manager brings in a mechanic to deal seconds and intentionally dump to an agent as part of a skimming operation. When the mechanic starts to worry and decides to bust out legitimate players in order to compensate for the amount of the skim, the casino manager tells him to stop doing so. The mechanic asks the CM; “How did you know I started dealing seconds?” The CM replies; “The numbers on your game changed.” Now as to whether this degree of detection is even possible, I won’t even speculate. But I thought it was a good example of how the big boys view games management: if the numbers are right, then things must be running right. If the numbers aren’t right, maybe they better start to look for reasons.

A leading expert in the field of volatility is a writer by the name of Alan Krigman. 

On his website, he has many interesting articles on volatility and the effects it has on gamblers.

Standard Deviation

“To gain an understanding about how much deviation from the theoretical house advantage can be expected, and what fluctuations can be considered normal (due to chance variation, a good place to start is a common statistical measure caked the standard deviation. The standard deviation tells how much variation can be expected when observing repeated occurrences of a random variable.”

PCM then goes on to describe how different gamblers making a series of one thousand $5 bets on red in roulette will lose different amounts, thus the amount a gambler wins or loses is considered to be a “random variable.” The standard deviation for the total amount won by the house will show whether these different outcomes will tend to be about the same or will be wildly different.

“A small standard deviation means the outcomes tend to be similar. A standard deviation of zero, the smallest possible value, would indicate the outcomes are always exactly the same. A large standard deviation indicates the outcomes are very different or highly variable. The standard deviation is a crucial tool in volatility analysis.

The next section shows how to calculate the standard deviation and use it to predict the likely maximum and minimum win that will occur in a series of wagers. It also shows how to determine whether an observed win or loss in a series of wager is merely due to normal statistical fluctuations or is more likely explained by other reasons.”

Wager Standard Deviation

“To access the likely deviation between actual win and house advantage in a series of wagers, computing the standard deviation for a single wager is first required. Since the standard deviation is the square root of measure known as the variance, defining the variance is a prerequisite to determining the standard deviation. The variance for a single wager can be calculated from the following formula:”

“The standard deviation is the square root of the variance:”

At this point PCM illustrates computing the wager standard deviation for a $5 bet on red on a double-zero roulette wheel by the use of a chart that I found lacking. It is for this reason that I will illustrate the problem by the use of steps:

Step One

What is the net pay for this bet? We will be using the variance formula for both of the possible results, or net pays. In this case the two net pays are +$5 and -$5. Since “x” is used to describe the outcome for the house, x = +$5 and x = -$5.

Step Two

What is the probability of each of these net pays? The probability of +$5 is 20/38 or .526316 and the probability of -$5 is 18/38 or .473684 So P_x = 0.526316 and P_x = 0.473684

Step Three

What is the expected value of this $5 bet on red? It is important to remember that they are using the expected value for the house and not the player, so this is a positive number. The house result will be +$5 about 53% of the time or:

+$5 * .526316 = 2.63158

The house result will be -$5 about 47% of the time or:

-$5 * .473684 = -2.36842

2.63158 – 2.36842 = 0.26316 = the EV of this $5 bet on red.

We now know the net payoffs for the bet (either +$5 or -$5) the probability of each payoff (.526313 and .473684 respectively) and the EV of the bet (.26316). So we have all the information needed to compute the variance and thus the wager standard deviation of this $5 bet on red.

Step Four

X - EV

Subtract the expected value (EV) from each of the results (x).

+$5 – .26316 = 4.73684

- $5 – .26316 = -5.26316 (I guess subtracting a negative number from a negative number is like adding them together).

Step Five

(X – EV)²

Square each of those amounts.

4.73684² = 22.43765319

-5.26316² = 27.70085319

Step Six

(X – EV)² * P_x

Multiply each of those amounts times the probability of them occurring.

22.43765319 * .526313 = 11.80922856

27.70085319 * .473684 = 13.12145094

Step Seven

Add those amounts together to compute the variance of the wager.

11.80922856 + 13.12145094 = 24.9306795 this is the variance for a $5 bet on red.

Step Eight

Compute the square root of the variance on your abacus.

Or use your calculator and arrive at:

4.993063138 this number is the wager standard deviation for our $5 bet on red.

What we are supposed to have learned from all this is that the expected value of a $5 bet on red is $.263 and represents how much the house can expect to win on every $5 bet on red. The standard deviation for this bet is $4.993. What this all means is still a mystery to me but we get a hint from the next paragraph:

“The standard deviation is the key to assessing the likely fluctuations that will occur in a long series of wagers. It is easiest to perform the necessary analysis using the expectation and deviation on a per unit basis.”

Per-Unit EV and Standard Deviation

“The per-unit expected value and standard deviation can be obtained by computing the wager expected value and standard deviation, as shown above, assuming a $1 bet, or from the wager expected value and standard deviation as follows:”

Quite simply, you divide the EV or SD for a wager by the wager amount. All of this crap is unnecessary if you just use the example of a $1 bet when you compute the EV or SD in the first place.

Using the above formulas for our $5 bet on red will produce the following results:

The per-unit EV of .052632 means the casino will keep 5.2632% of every bet, regardless of amount, in the long run. And no, you haven’t missed anything if you don’t know what to do with this SD of .998614. My guess at this point is that since the SD is approximately 20 times the amount of the EV, if a flat bettor (a gambler that always bets the same amount) were to be winning or losing 20 units, this would fall under one standard deviation.

Alternative Formula for Variance and Standard Deviation

I really don’t think we need any more formulas but since this one is shown in PCM, it will be shown here.

And so, here it is in action on our $5 bet on red:

VAR = [(+5)² (.523316) + (-5)² (.473684)] – (.263158)²

= 25 - .069252

= 24.930748

And of course, the SD is computed by determining the square root of the variance.

Coming Soon!

Confidence Limits

© 2005 Dale S. Yeazel

www.CasinoDealers.net

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