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Probability

The theory of probability is an entertaining branch of mathematics that has occasionally been the subject of controversy.

In the beginning, the notion of probability was considered by some to be impious; everything that could either happen, or fail to happen, will do so in accordance with God's will, and thus the concept of probability is meaningless.

And, of course, the clearest classical illustrations of probability are associated with the disreputable activity of gambling.

But other controversies are also associated with probability. Bayesian statistics are a recent invention whose validity is controversial.

What are Bayesian statistics?

In classical probability, one starts with assumed probabilities: a die has six identical sides, and so we assume that an ideal die has an equal probability to have any of its six faces uppermost after being thrown. From that, we can then draw conclusions about the chances of various combinations of dice being thrown.

Bayesian statistics, instead, deal with inferring probabilities by induction. If we observe an event happening on the average one time out of every ten, we can conclude that its actual probability of happening is most likely one time out of ten, and, depending on how many observations we have made, we can also talk about the probability that the real probability is higher or lower than the observed value.

But in a physical system, an actual probability of an event happening already exists, and all that can happen is that during a series of observations, one might, with a certain probability, see more or fewer occurences of that event than are more likely. In a sense, this simply takes the original objection to probability, based on a deterministic universe, to the next higher level.

The response, of course, is that just as the roll of a die is deterministic if we could know the state of every atom in the air through which the die is thrown, and the brain and body of the thrower, actual probabilities of events are things we may either know or be ignorant about, and in the absence of an opportunity to obtain knowledge of a priori probabilities in a reductionist fashion, by breaking down a system into its component parts, each of which are simple enough to fully understand, observation of behavior is another method to obtain information.

The tragedy of the Space Shuttle Columbia provided an illustration of the meaning of Bayesian statistics. In a news report after that tragedy, Jesco von Putkammer was interviewed by a reporter, and he denied that the tragedy meant that the Space Shuttle was necessarily unsafe. In effect, he noted that if the chance of a fatal accident with the Space Shuttle were one in a million, that would not prevent the accident from taking place on the 25th flight, as it did, instead of the 1,000,000th flight, or the 500,000th flight.

And that is absolutely true, yet it is also reasonable, in the absence of other information, to conclude that the likeliest value for the risk is on the order of one in fifty, even if that value is itself highly uncertain.

A Trivial Example

Much of classical probability theory would seem to be nothing more than an exercise in simple arithmetic.

Thus, although it is complicated, there seems to be nothing profound about calculating the odds for a slot machine or one-armed bandit. This is a gambling device that has three reels, each with 20 symbols on it, these symbols often being chosen from a set of six or seven different symbols.

One example of the assortment of symbols that can appear on a slot machine is:

(*) Lemons       3   -   4
(%) Cherries     7   7   -
(O) Oranges      3   6   7
(@) Plums        5   1   5
(A) Bells        1   3   3
(=) Bars         1   3   1

which is designed for a machine with the following schedule of payouts:

%%.        3
%%*, %%A   5
OOO, OO=  10
@@@, @@=  14
AAA, AA=  18
===      120

This payout schedule is one of those illustrated below.

A large number of slot machines made prior to World War II by many manufacturers had the schedule of payments shown above, except that the combination of three bars paid out 20 coins plus the jackpot, whatever it might be; for purposes of a simple calculation, a fixed jackpot of 100 coins is assumed.

The number of times the various combinations of symbols would occur, on average, if the machine were operated 8000 times can be calculated in a straightforwards manner, and these probabilities (scaled up by a factor of 8000, the total number of combinations) can then be multiplied by the amount paid for the combination to work out how profitable the machine is:

Reel           Combinations  Value   Amount
 1    2    3                         Paid

 7 *  7 * 13 =          637  *   3 =  1911
 7 *  7 *  7 =          243  *   5 =  1215
 3 *  6 *  8 =          144  *  10 =  1440
 5 *  1 *  6 =           30  *  14 =   420
 1 *  3 *  4 =           12  *  20 =   240
 1 *  3 *  1 =            3  * 120 =   360
                                      ----
                                      5586

for a profit of 30.175%. Thus, in the first line, the calculation is based on the fact that there are seven cherry symbols on the first reel, seven cherry symbols on the second reel, and thirteen symbols that are neither a lemon nor a bell on the third reel.

More about the fascinating world of slot machines continues on this page.

A Less Trivial Example

The study of probability as a mathematical discipline can be said to have its start in a question concerning the throw of three dice. A mathematician was asked why it should be likelier to throw 10 with three dice than to throw 9. Each combination can be made in six ways:

9 can be made as 1-2-6, 1-3-5, 1-4-4, 2-2-5, 2-3-4, 3-3-3;

10 can be made as 1-3-6, 1-4-5, 2-2-6, 2-3-5, 2-4-4, 3-3-4.

Although there are 56 visibly different combinations that can be thrown with three identical dice, assuming that each one is equally likely is a mistake. Realizing this is what began the journey to making probability a science.

It is reasonable to assume that one die is equally likely to come up with any one of its six faces. Starting with what we do know, instead of guessing what three dice might do, to break down their behavior into a combination of the behaviors of three dice, one thinks of three dice that differ in color, or which are thrown in three separate dice cages.

This then leads to the conclusion that instead of 56 equally probable outcomes, there are 216 possibilities equal in likelihood. A combination where all three dice come up with the same number, such as 3-3-3, reflects only one of those possibilities, but one where all three dice are different, such as 2-3-4, reflects six of them. When two of the dice show the same number, and the third one is different, three of the 216 possibilities are involved.

Thus, the chance of throwing a 9 is 6+6+3+3+6+1 or 25 out of 216, while the chance of throwing a 10 is 6+6+3+6+3+3 or 27 out of 216.

The odds of throwing different totals with three dice approximates a bell curve or normal distribution, and this has made the use of three dice popular for determining traits in role-playing games:

   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18
   1   3   6  10  15  21  25  27  27  25  21  15  10   6   3   1

A Historical Example

Most people are familiar with the game of Craps.

In this game, the player throwing a pair of dice wins if he throws 7 or 11 on his first throw, but loses if he throws 2, 3, or 12 on his first throw. If he throws another number, then he continues to throw the dice until he either throws that number again and wins, or he throws a 7 and loses.

The chance of throwing each of the possible numbers from 2 to 12 with two dice follows a triangular distribution:

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

and the chance that the player throwing the dice will win can be determined by looking at the thirty-six possibilities for the first throw and their consequences:

 2 is thrown 1 time, and leads to a chance to win of 0
 3 is thrown 2 times, and leads to a chance to win of 0
 4 is thrown 3 times, and leads to a chance to win of 3/9
 5 is thrown 4 times, and leads to a chance to win of 4/10
 6 is thrown 5 times, and leads to a chance to win of 5/11
 7 is thrown 6 times, and leads to a chance to win of 1
 8 is thrown 5 times, and leads to a chance to win of 5/11
 9 is thrown 4 times, and leads to a chance to win of 4/10
10 is thrown 3 times, and leads to a chance to win of 3/9
11 is thrown 2 times, and leads to a chance to win of 1
12 is thrown 1 time, and leads to a chance to win of 0

The reason the chance to win is 3/9 when 4 is thrown is because there are 3 ways to throw a 4, 6 ways to throw a 7, and only those two throws count, and the first one is decisive, afterwards.

The chance to win, therefore, is:

1   3   2   2    5    5   1    1
- * - + - * - + -- * -- + - + --
6   9   9   5   18   11   6   18

which works out to 244/495, or 49.292929... percent.

This might seem complicated enough. But the game of Craps is a simplified descendant of the game of Hazard.

In the original form of the game of Hazard, the two dice are thrown first to establish the main point; this can only be 5, 6, 7, 8 or 9, and they are thrown again until such a number is obtained.

Then, they are thrown once to obtain a chance point; this can be 4, 5, 6, 7, 8, 9, or 10. If, on that throw, the main point is thrown again, the caster wins. As for the throws of 2, 3, 11, and 12, their significance depends on what the main point is:

If the main point is:            Wins    Loses
 5 or 9                                   2, 3, 11, 12
 6 or 8                           12      2, 3, 11
 7                                11      2, 3, 12

When the main point is 7, which it will be 6 times out of 24 (or 1 time out of 4), one obtains the same probabilities of winning as in the modern game of Craps.

When the main point is 6, the chances of winning are as follows:

 2 is thrown 1 time, and leads to a chance to win of 0
 3 is thrown 2 times, and leads to a chance to win of 0
 4 is thrown 3 times, and leads to a chance to win of 3/8
 5 is thrown 4 times, and leads to a chance to win of 4/9
 6 is thrown 5 times, and leads to a chance to win of 1
 7 is thrown 6 times, and leads to a chance to win of 6/11
 8 is thrown 5 times, and leads to a chance to win of 5/10
 9 is thrown 4 times, and leads to a chance to win of 4/9
10 is thrown 3 times, and leads to a chance to win of 3/8
11 is thrown 2 times, and leads to a chance to win of 0
12 is thrown 1 time, and leads to a chance to win of 1

and the chance of winning is the same when the main point is 8; the roles of the throws of 6 and 8 are interchanged, but as their likelihood is identical, the overall chance is the same.

The chance of the main point being either 6 or 8 is 10 times out of 24 (or 5 times out of 12). When the main point is either 6 or 8, the caster's chance of winning is 6961/14256, or 48.82856341189674523007856341189674523007856... percent.

When the main point is 5, the chances of winning are as follows:

 2 is thrown 1 time, and leads to a chance to win of 0
 3 is thrown 2 times, and leads to a chance to win of 0
 4 is thrown 3 times, and leads to a chance to win of 3/7
 5 is thrown 4 times, and leads to a chance to win of 1
 6 is thrown 5 times, and leads to a chance to win of 5/9
 7 is thrown 6 times, and leads to a chance to win of 6/10
 8 is thrown 5 times, and leads to a chance to win of 5/9
 9 is thrown 4 times, and leads to a chance to win of 4/8
10 is thrown 3 times, and leads to a chance to win of 3/7
11 is thrown 2 times, and leads to a chance to win of 0
12 is thrown 1 time, and leads to a chance to win of 0

and the chance of winning is the same when the main point is 8; the roles of the throws of 5 and 9 are interchanged, but as their likelihood is identical, the overall chance is the same.

The chance of the main point being either 5 or 9 is 8 times out of 24 (or 1 time out of 3). When the main point is either 5 or 9, the caster's chance of winning is 1396/2835, or 49.241622574955908289241622574955908289... percent.

Given the probabilities of the different main points, the caster's chance of winning in the original form of the game of Hazard is 1979/4032, or 49.0823412698412698... percent.

As the probabilities of winning for each of the possible values of the main point are so nicely balanced, to speed things up, the rule was changed so that instead of first rolling the dice until a main point was obtained, the caster could simply choose, and announce, his own main point before throwing for the chance point.

Since the caster's chance of winning was 49.29% for a main point of 7, 49.24% for a main point of 5 or 9, and 48.83% for a main point of 6 or 8, 7 was the caster's best choice, which led to the modified game of Hazard quickly changing to the modern game of Craps.

In the form of Craps played in a casino, Bank Craps, people can also bet against the player throwing the dice; so that this bet is also profitable for the casino, the most common rule is that if the caster loses by throwing a 2 on the first throw, a bet against the caster is not lost, but does not win either. This makes the house percentage on a bet against the caster only slightly less than that on a bet for the caster. Some Bank Craps layouts instead bar 3, making the percentage on a bet against the caster considerably higher; others offer bettors a choice of betting against the caster with 2 barred, or of betting against the caster with 12 barred; this provides a small additional reassurance of fair play.

In Hazard, since the caster's chance of winning is significantly smaller for a main point of 6 or 8, if 2 were barred on a bet against the caster, although the house would have an advantage, it would be only 0.435 percent, between a quarter and a third of that for the other bets with or against the caster; thus, either 3 or 11 (which is a losing throw when 6 or 8 is the main point, 12 in this case being a winning throw) would more likely be barred were there such a thing as Bank Hazard.

And if there were, the betting layout might look like this:

The Big 6 and Big 8, and 4, 6, 8 and 10 the hard way would always be rolled against 7, as in Craps, so the odds of these bets would remain the same; even money for the Big 6 and Big 8, 7 to 1 (or 8 for 1) for 4 and 10 the hard way, 9 to 1 (or 10 for 1) for 5 and 9 the hard way.

The layout is shown here with odds that give a lower house percentage than is usual on a few bets. Thus, a field bet pays triple, rather than double, if 2 is rolled, and bets on 2 or 12 pay 31 to 1 (or 32 for 1).

The rule of play for Bank Hazard that is envisaged as being applicable to this layout is that the shooter may announce his main point, whether 5, 6, 7, 8, or 9, and may also announce "Roll for Main" or just "Main" to indicate that his main point will be decided by the dice; if no announcement is made, the main point is 7 by default; thus, unless a main point is specified, play proceeds as at Craps.

The shooter may also announce "Wrong" or "To Lose" or "Miseré" in addition to his point; in that case, he retains control of the dice if he throws a losing combination, including one that is barred (one that neither wins nor loses for wrong bettors) and loses control of the dice if he throws a winning combination. This does not affect which dice rolls win for bettors on the Pass Line and the Don't Pass line, thus, in that case, bettors who believe the shooter is "on a roll" would bet on the Don't Pass line in order to win.

Furthermore, when the shooter is rolling for a main point, instead of rolling the dice indefinitely until a number from 5 through 9 is thrown, it would seem appropriate for the rules to provide that a throw of 2, 3, 4, 10, 11, or 12 when rolling for the main point would make the default value of 7 the main point. This way, the main point is always decided by a single throw of the dice, removing what was perhaps the single biggest reason that Hazard was displaced by Craps.

The rolls which lose or win immediately on the throw for the Chance point are given above in the rules for Hazard.

The fact that each of the "Come" and "Don't Come" lines on the layout are divided into five areas, the feature which most strongly distinguishes it from a Bank Craps layout, gives the Come and Don't Come bettor the same freedom as the shooter to chose a point. Bets placed on the "Come to Main" and "Don't Come to Main" are moved to the appropriate space on the Come or Don't Come line below them when a roll from 5 to 7 decides the main point applicable to them.

Note that the Place bets on the layout include the number 7, unlike those on a Bank Craps layout. The meaning of these bets, therefore, is determined by the current main point of the shooter, and bets cannot be placed on the space corresponding to that point, which is covered by a marker. Since a bet can be placed on chance points other than the shooter's chance point, if the shooter wins, it is possible that a bet might be undecided at that time, and the shooter can call a different main point for his next throw (or the point can change if he is remaining with "Roll for Main"); hence, there are two rows above the place area to which bets can be moved that remain associated with a different main point no longer in use, that point being indicated by a marker of appropriate size.

The odds for the place bets are:

      Main Point is 7               Main Point is 6 or 8           Main Point is 5 or 9

      True     Normal    Normal     True     Normal    Normal      True     Normal    Normal
                         Wrong                         Wrong                          Wrong

4,10  2 to 1   9 to 5    5 to 11    5 to 3   6 to 4    2 to 4      4 to 3   5 to 4    2 to 3
5,9   3 to 2   7 to 5    3 to 5     5 to 4   6 to 5    3 to 6      1 to 1   11 to 12  11 to 12
6,8   6 to 5   7 to 6    4 to 5     1 to 1   11 to 12  11 to 12    4 to 5   3 to 4    6 to 5
7     --       --        --         5 to 6   4 to 5    7 to 6      2 to 3   7 to 5    3 to 5

A place bet or don't place bet at true odds may be made up to what one has wagered on the pass line or the don't pass line.

The spaces for Under and Over 7 are restored to the layout, but they are modified so that Under 7 also pays when 7 is rolled as 1-6, and Over 7 also pays when 7 is rolled as 3-4, so that the percentage on those spaces is reduced to 5 5/9%. Also, in addition to a space for betting on 2, 3, and 12, the three combinations which are craps when 7 is the main point, spaces are provided for betting on 2, 3, and 11, the three combinations which are craps when 6 or 8 is the main point, and for betting on 2, 3, 11, and 12, all four of which are craps when 5 or 9 is the main point.

Normalized Percentages

Many books on gambling will note that, in an American casino, a bet for the caster at Craps carries a house percentage of 1.414% against it, while a bet on a single number at Roulette, on the American layout with two zeroes, carries a house percentage of 5.263% against it, as does a bet on red or black, since American roulette does not have the prison rule of European Roulette, with one zero, which reduces the percentage against these bets to half that against that of other bets on the layout.

It is quite correct, therefore, that a bet on the line at Craps would be a better bet than a bet on red or black, or odd or even, or high or low, in an American-style casino.

But let us suppose one has a single chip in one's hand, and one's intent is to gamble it so as to win about 30 times its value if one is lucky.

One can bet it on a single number at Roulette, and one's average return will be 5.263% less than the chip's value, although what will actually happen is that one will either lose it, or win 36 times its value. Or one can let it ride five times in a row at Craps, to either lose it, or win 32 times its value; then, one faces a percentage of 6.8735% against one.

To compare different bets offered in a casino in a useful manner, as instruments towards a goal of winning one's stake times a given amount, a game with a probabity p of winning n times one's stake can be considered as equivalent to a game with a probability p^(log(N)/log(n)) of winning N times one's stake, and different games should be compared with a constant N. How to apply this rule to games with multiple different payouts, such as playing the field in Craps, or gambling with a slot machine, however, is less obvious; presumably, the best way is to divide the input stake between the possible payouts in such a way that the normalized expectation for each one is identical.

Arranging Numbers

If one wishes to talk about mathematics and gambling, another item of interest is the arrangement of numbers on the roulette wheel:

the diagram above shows how the numbers on the European and American roulette wheels are arranged to allow red and black, high and low, even and odd, the three dozens and the three columns all, simultaneously, to come close to alternating.

The following diagram

illustrates the difficulties faced in making an improved order of numbers for roulette wheels. In the first half of the top row, we see ten numbers consisting of five that are red, odd, and low, and five that are black, even, and high.

Alternating numbers from those two groups, to give a stretch of ten numbers that alternate from red to black, from odd to even, and from low to high, while it could lead to a good mixture of the numbers from the three columns, would lead to the result containing five numbers from the first dozen, three numbers from the second dozen, and two numbers from the third dozen.

If priority is given, as on the European roulette wheel, to the alternation from low to high over that from odd to even, the eight numbers in the first half of the bottom row that consist of four that are red, even, and low, and four that are black, odd, and high will complement this first group in respect of the distribution of the three dozens.

The following diagram shows one possible improved ordering for the numbers on the roulette wheel in the top row:

the second row shows an early French arrangement of the numbers on the roulette wheel that appears in at least one old engraving of a roulette wheel and layout, and the bottom row shows an arrangement that appeared in many old books on the rules of various games.

In the two older arrangements shown, the numbers that are red and black do not alternate.

I believe that what had been the case for roulette wheels laid out according to these two older arrangments is that the numbers on the wheel still did alternate between red and black, since bets on the red and black colors were still provided for both of them, according to the sources for those wheel arrangements, but which numbers were red, and which numbers are black, differed from the modern case.

The earliest roulette wheels in France, like those currently in use in United States casinos, had both a zero and a double-zero. However, some sources note that instead of being green, 0 was red - and it was considered to be odd and low as well; and 00 was black - and it was considered to be even and high as well.

This did not mean that the even-money bets gave the house no percentage; if a zero or double-zero won for such a bet, it neither lost nor won; thus, for those bets, it was as if the wheel only had one zero instead of two.

This, I think, explains why the modern European single-zero roulette game includes the prison rule; when the single-zero roulette wheel was introduced in 1843 by Louis and François Blanc at Bad Homburg, in order to be accepted as a true improvement on the existing double-zero roulette wheels, it needed to cut the percentage in half on all the bets, including the even-money bets, which, at the time, had a house advantage as if only one zero were present on those double-zero wheels.

The prison rule in European roulette is this: when zero comes up, the even-money bets, instead of being lost outright, are moved to an area within the spaces on which they are placed, called the prison, and then on the next spin, either they are lost, or, if this time they win, then they are simply returned (to the main area of the same space of the layout on which they were placed, from which the player may retrieve them) without additional winnings; once again, the same thing as is called "no action" in Craps.

Later on, since coloring 0 red and 00 black caused confusion and complaints, those numbers were colored green instead, and both zeroes simply lost on the even money bets as is the case today with the 38-number wheel.

Alternation of the major bets on the roulette layout, however, is not the only characteristic that an arrangement of numbers on the wheel should satisfy. Various bets on three, four, or six numbers are possible on the roulette layout as well. In order to ensure that the numbers involved in any such wager are widely distributed on the wheel, it is sufficient to ensure that the numbers in any two consecutive rows of the layout are widely distributed. The following diagram:

illustrates how the arrangements considered above fare in this respect.

The Old American Roulette

While the Roulette wheel with both a 0 and a 00 originated in France, even back when it was common there, a higher percentage was taken on Roulette wheels in the United States, at least some of the time.

Several old books on games pictured an American roulette game where the wheel had the numbers from 1 to 28 instead of 1 to 36, and in addition to 0 and 00, a third zero was present, represented by the image of an Eagle.

These books showed the following arrangement of numbers on those wheels:

 E 20 11 18 13 16 27  2 25
 0  4 23  6 21  8 19 10 17
00 12 15 14  1 28  3 20  5 24  7 22  9

Spaces on the layout are also shown for betting on red or black, but not for the other even-money bets. The numbers from 1 to 28 are organized in four columns, and a bet on a single column is also possible.

Which of those numbers are red, and which are black, though, is something at which I could only guess. However, recently I saw a photograph of a roulette wheel of this general type which was offered at auction. But it had a different arrangement of numbers on it:

 E 16  3 12 21  4 25  8 17 22
    R  B  R  B  R  B  R  B  R

 0 14  7 26 15 10 19  2 23  6 27
    R  B  R  B  R  B  R  B  R  B

00 13 20  9 28 11 18  5 24  1
    B  R  B  R  B  R  B  R  B

thus, the three zeroes are spaced as equally as is possible, the 0 is flanked by two red numbers, and the 00 is flanked by two black numbers.

And so all the odd numbers are black and all the red numbers are even on this wheel, which explains at least why there are no Even or Odd bets on the layout, if not the absence of High and Low bets.

Odd and even numbers also strictly alternate in the arrangement shown for 31-number wheels in the game books, except that in that arrangement the 0 and 00 are also flanked by one odd and one even number, like the Eagle, and so it is very likely that the same convention applied to them.


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