Distributions of Random Variables in Gambling

The little ball in roulette will land on a predetermined number 2.63% percent of the time. The odds are 37:1 against. These are just different ways of saying that there is one chance in 38 that the “desired” outcome will occur. It would not be hard to spin the roulette wheel a few thousand times and record the results (but it would be boring, to be sure!). While some numbers will almost certainly come up more often than others (“random variation”), the frequency of “hits” around a true wheel will tend towards 2.63% in each.

Think of rolling a fair die. Before you throw it, I ask you, “What are the chances that it will be a ‘six’?” You answer intuitively, “One in six, as the die has six faces, and the ‘six’ is one of them.” You will be right. Now roll the die. It comes up a “five.”

Then I ask you, “What are the chances that when you roll it a second time it will be a ‘six’?” Your answer should not change. The first throw is independent of the probabilities of any later throws.
If you roll it again, and it comes up a “five” for a second time, you may shrug your shoulders and ask, “What are the chances?” The answer is “1 in 36 that you would roll two consecutive ‘fives.’”
That does not mean that it was a bad bet to bet on a “six.” Had you bet, the correct question would have been to ask how much you paid and how much you could have won. The out-come does not validate the bet. The prospect does.

The two previous examples are of integer outcomes (a six, a 27, a win or a loss in sports or racing). They deal with the distribution of a countable number of outcomes over a range of possibilities, and are most common in gaming. But they are just special cases of a more general conclusion concerning random events.
To illustrate the more generalized truth, consider something not related to gaming, say, the height of Las Vegas sixth graders. This is a continuous, infinitely divisible variable, subject to random variation. We could have also used the distribution of yards gained in a football game or some other variable that is a continuous function.

If the heights of all the Las Vegas sixth graders were arrayed in graph form from shortest to tallest, the graph would probably look like the familiar bell curve. It might not be symmetrical, but it would have a high point (the mean or average), and it would disclose some variation or dispersion around the mean.
If the compilation were to take into account a couple of variables that we know to be not random influences (like gender, age, height of parents, ethnicity perhaps), the resulting, adjusted graph, would be more symmetrical and more like the bell curve than the first one. If we somehow were able to account for all non-random influences, the remaining graph, representing only random variation, would be precisely the “normal” distribution.

Figure 1 is a picture of this distribution, indicating the “normal” dispersion of the variable around the average (conventionally indicated by the Greek letter mu (?)), and subject to a standard deviation (conventionally indicated by the Greek letter sigma (?)). To use gamer’s lingo, the “odds” that an observation of the variable described by this probability distribution will fall within a standard deviation of its average, on either side, are 68.2% or a little better than 2:1.

So, returning to gaming, the examples of the roulette wheel and the dice do illustrate that generating events repeatedly through a random process, over and over, will generate a distribution like the bell curve (or a relative of it) with the peak at the “mean” (the average, or the expected value) and the remaining observations falling off as they are more distant from the mean (as measured by the distribution’s variance, or the square root of the variance, called the standard deviation).

In the case of the roulette wheel, the different experiments are said to be arrayed in a “distribution” with an average of 2.63%, with random dispersions on either side of the 2.63%. The incidence of values far away from the mean will be very small. In fact, the distribution of roulette experiments will also look like the familiar “bell” curve, with a mean of 2.63% and a “variance” such that 95% of all the observations will be “very close” to 2.63 if not precisely on it. How close is “very close” depends on the standard deviation of the distribution. In the case of the normal distribution and its relatives that would apply to the roulette wheel, 95% of the outcomes are expected to fall within two standard deviations on either side of the mean, and 99.7% will fall within three.

The mean, the variance (and its square root, the standard deviation) are “statistics,” which means that they are mathematical measurements (for which there are formulae) of the distribution itself.

The Law of Large Numbers is a probability theorem that pertains to all random variables, including the heights of Las Vegas sixth graders and also the outcome of roulette wheel spins and throws of a die. When experiments are performed (or “samples” are taken), the outcomes will naturally portray an underlying distribution of outcomes, called a sampling distribution. These isolated points are taken as measures of the sampling distribution, that is, the “true” distribution of the random variable in the sample, perceived as if the experiment were performed an infinite number of times.

The sample is an attempt to discover the “sampling distribution.” Think of the sampling distribution as the complete picture or paradigm of what the “true” sample would be like. The finite number of observations would be dots on the map. The “dots” are a reflection, a hint, at what the researcher would obtain if the experiment went on infinitely, to paint the complete picture. This certainly saves a lot of time for researchers! Likewise, to obtain a clear picture of heights of Las Vegas sixth graders, one need not measure each one. A large enough random sample would be sufficient.

Most importantly, the Law of Large Numbers says something about the underlying “thing” being measured, that is, the distribution of the outcomes in the real world, the “population” and not just the “sample.” Not only is the sample a fair sketch of the distribution for the sampling experiment, it is also a fair sketch of what would happen with all fair roulette wheels in the world. The mean (average) of the sample population is predictably close to the average of the underlying reality. And the same is true for the variance of the distribution. (Remember, the variance is a measure of how clustered the observations are around their mean).

Thus, the Law of Large Numbers permits us to conclude that any true roulette wheel tends to cough up a 27 (or whatever) 2.63% of the time.

So the first principle is that certain events – those with random variation -- have probabilities attached to them, which means that their outcomes can be precisely predicted when the events are repeated a large number of times. This permits us to think of such events as having “expected values.” In this text they are referred to as “physical expectations.”

Suppose now, as is the case in most Las Vegas gaming, that the question is not something measurable, like the height of children, but rather a “yes-no” experiment or an experiment in which there are a small number of integer outcomes.

Binomial Distribution

The first of these special cases relies on the binomial distribution, a relative of the normal distribution. It tells us the probability of a number of successes out of a number of trials. For example, throwing the die 20 times, looking for a six. By now, you will understand that the probability of a six on a given throw is 1 in 6. What is the expectation of sixes in 20 throws? The binomial distribution for n=20 and p=.167 yields an average (expected value) of 3.33. This can be calculated from a formula, or looked up in a table [link].

The expected value or mean of a binomial distribution is n * p. The variance is the mean times (1-p), in the case of the die experiment, 2.78. (The square root of the variance, 1.66, is the common measure of the dispersion of the distribution around the mean). Figure 2 is a picture of a binomial distribution, showing the shape of one where the number of tries is 20 and another where the number of tries is 40. In both cases, the probability of a successful outcome on a given try is .5, as in the case of a coin toss.

The requirements for the binomial distribution are: (a) that the outcomes of each trial must be “binomial,” that is, either yes or no, on or off, 0 or 1, true or false; (b) that the outcomes be independent of one another; and that the probability be the same for each trial. (Each trial is sometimes called a Bernoulli trial, after the mathematician).

Poisson Distribution

The binomial distribution is less useful when the probabilities become very low, the so-called “rare event.” Fortunately, if the number of trials is reasonably high, the distribution becomes clearly defined and can be used in the same way as a binomial distribution. A simple example is the throwing of a die, with a .167 chance of a given outcome on each throw. How often can one expect to see, say, more than 5 sixes in a given experiment in which the die is thrown 20 times?

The French mathematician Simeon Denis Poisson identified how the binomial distribution trends (or in modern terms, “morphs”) as the number of trials goes towards infinity, while the mean of the distribution (n*p) stays where it is (i.e., with a corresponding reduction in the probability). This distribution is extremely useful when probabilities are very small, which is the case a lot in Las Vegas betting propositions.

Once a bettor has the p figure, a Poisson table can tell what the probability is of achieving some other outcome, given (implicitly) an “n” that is very large (perhaps 20 or more). For example, in the case of the die, where the mean of the distribution is 3.3, the probability of having exactly 3 sixes is 22%, 4 sixes is 18%, and 2 sixes is 20%. This means that 60% of the probability is centered on outcomes of 2, 3 or 4 sixes out of 20. There is virtually no chance of 9 or more sixes when the expected value is 3.3% (effectively zero). To answer the precise question put in the example, the chances are only 11% of getting more than 5 sixes out of 20 throws. Figure 3 is a picture of the Poisson distribution.

Now suppose the betting proposition is something like “over 6 incomplete passes” from a specific quarterback in an upcoming game. By research and analysis, you conclude that the expected number of incomplete passes per game for this quarterback, adjusted for game conditions and opposition, is 8. What are the odds that in this coming game the quarterback will generate 7 or more incomplete passes? (Note, the proposition is “over 6,” meaning “6 or fewer” is the opposite side of the bet.) A Poisson table, like the sample in Figure 4, shows (if you add the odds for 0-6) a 31% chance of as many as six, when the mean is 8. This means that the odds of 7 or more incomplete passes, given a mean of 8, is 69%. (Many Poisson tables are available in cumulative format, to avoid having to add up the rows.) For extra credit, calculate your edge on this proposition, assuming the offered terms were 6.5 incomplete passes at -110.

For the Poisson Distribution to be useful to the gamer, a couple of requirements must be met: (a) The outcomes must be countable, that is, integers, counted one at a time, rather than analog, quantitative measurements like yardage or the heights of school children; (b) the expected number of occurrences is proportional to the length of an interval, such as 20 throws, a football game, a ten-minute span; and (c) the probability of an occurrence on a given trial is fairly low, but there are a large number of trials. In the football example, the number of incomplete passes is countable. It’s not a quantity, like yardage. The number of incompletes is judged to be proportional to the length of the game, and this is standard from one observation to another. (Otherwise the outcomes would not be comparable, or would have to be adjusted to reflect the same interval). Finally, the odds of an incomplete pass are fairly low on any given play, but there are many plays in a game.

The horizontal axis is the index k. The function is non-zero only at integer values of k. The connecting lines are only guides for the eye and do not indicate continuity.