Serious scientific investigation of the laws of chance (which came to be known as mathematical Theory of Probability) seems to have begun in Renaissance times, when so many other pioneering ventures in science were taking place.  And some of the early investigations were made by the same great thinkers who were undermining age old superstitions and paving the way for modern science men like Johannes Kepler or Galileo Galilei.

Kepler, as an astronomer, was concerned with stars, not Gambling; but when a bright new star appeared in 1604, he collected the views of some other stargazers and approved the theory that the star had appeared because of the chance concurrence of atoms.  Thereupon, he made a stab at some calculations in order to determine the time and the mathematical probability of another similar concurrence.

Galileo’s contribution to the probability theory has a more direct relationship with gambling itself.  He turned aside from his impressive work in other scientific fields to answer the trivial query of a gambling friend.  The friend wanted to know why, with three dice, the number Ten is thrown more often than the number Nine.  Galileo prepared an analysis of chances and showed that out of 216 possible cases, the number Ten has the gaming advantage over the number Nine in the ratio of 27 cases to 25, because there are 27 combinations of dice forming the number Ten and only forming Nine.

On an English race course white-gloved “tic-tac” men use a complex code of hand signs to signal the latest odds to bookies around the course.  Racing odds are not mathematically fixed; they depend largely on a horse’s past “form,” and fluctuate as pre-race bets are made.

But the man who made the most extensive early examinations of probability was Gerolamo Cardano (1501-76), sometimes called Cardan.  While a student at University of Padua, Cardan began assembling notes for his Libber de Ludo Aleae (the Book of Games of Chance).  He had plenty of opportunity for studying the subject, for his income was mainly derived from gambling until he achieved some fame as a physician, mathematician, philosopher, and inventor.

The Book of Games of Chance is a scrappy compilation; Cardan would frequently work out a solution to a problem, later discover that he was in error, and confusingly leave both the wrong and the right answers without any reference to the links between them.  But it is comprehensive in subject matter, if not in treatment.  Moral, historical, practical, and arithmetical aspects of gambling are all considered-though some of them not very deeply.  Cardan warns his readers that, if they must gamble, they had better gamble for small stakes and that their opponents “should be of suitable station in life.”  He adds, however, that “in times of great anxiety and grief [gambling] is considered to be not only allowable, but even beneficial.”  There are instr5uctions for playing primero (a card game similar to poker ) and hints on watching for cheats who use soapy cards and mirrors in their finger rings to reflect the playing surfaces.

Aside from all his material the book includes Cardan’s notes on the principles of probability.  He began his cogitations logically enough by considering that a die has six sides and that in a single cast of the die (since there is no skill or factor other than chance involved ) any one of the sides is as likely to fall upper most as any other.  “Six equally likely cases” was Cardan’s actual phrase.  The probability of a particular side of the die falling uppermost he therefore expressed by the fraction 1/6.

Thus, for determining the probability of  an event that is governed by pure chance, , the universal formula is p = f/c (p being the probability, c being the total number of possible cases, and f the total number of favorable cases).  Applied to the tossing of a coin, the fraction would be ½, since there are two sides to the coin and one chance in a single toss that either the head or the tail will fall uppermost.  To make this clear to the mind that is always dazed by formulae, here is the first Law of Probability in words.

The probability of an event is the lucky number of cases favorable to that event compared with the total number of possible cases, so long as all the possible cases are equally likely to happen.


Cardan then went on to calculate the probabilities with two and three dice.  He saw that with a single throw of two dice the total number of possible cases is 36, because any one of the six sides of one die can appear in combination with any one of the six sides of the other ( 6 x 6 = 36).  And with a single throw of three dice yet another six sides have to be taken into account, so that the sum would be 6 x 6 x 6 = 216 possible cases.

The calculation of possible cases is a simple matter of multiplication; but the calculation of favorable cases when two or more dice are being thrown is more difficult.  Cardan first took considerable trouble to work out all the possible ways in which a dice player can throw a specific number.  With this useful knowledge any player can work out his chances (or “favorable cases” ) of throwing a particular number in any single throw of, say, two dice.  It becomes clear that though he has only one chance of throwing a Two or a Twelve, he has six chances of throwing a seven.  Fractionally expressed as favorable cases, these are 1/36 and 6/36 (6/36 is expressible also as 1/6).  Not only that.  The dice player can now tell exactly what the odds are against his throwing any particular number, and from that knowledge can decide the amount of his bet or whether he should bet at all The odds can be determined by simply comparing the unfavorable cases with the favorable ones. 

In throwing a single die, for instance, there are six possible cases- the six sides of the die.  You therefore have a one in five chance of throwing a specific number with one.  Dies in one throw.  Similarly, with two dice there are 36 possible cases.  A Twelve or a Two can each be thrown in only one way; so the odds against your throwing a Twelve or a Two in one throw of two dice are 35 to 1.  The odds against scoring Eleven (or any other number that can be thrown in two ways) are 17 to 1 (i.e., 34 to 2).  In tossing a coin, the odds are equal, for it is equally likely that heads or tails will fall uppermost.  In drawing a card from a 52-card pack, the odds against its being the king of Clubs (or any other specified card) are 51 to 1.  On a 38- number roulette wheel, the odds against any one number’s coming up are 37 to 1.

With Cardan’s information about the favorable chances of throwing a particular number, you can also work out the odds against throwing, say, a Six before a Seven.  Such information will come in very handy if you play craps, a game (developed by American Negroes in the early 19th century ) that is a simplification of a European dice game called hazard.  Craps is played with two dice, and the rules are simple.  The thrower of the dice wins if on his first try he throws a Seven or an Eleven (a “natural” or “pass”).  He loses if he throws a Two, Three or Twelve (a “crap” or “miss-out”), but he may continue to throw.  If he throws a Four, Five, Six, Eight, Nine, or Ten (which is called the thrower’s “point”), he neither wins nor loses but goes on throwing the dice until he either duplicates his point (also a pass or win ) or throws a Seven (a miss-out of loss).

            It is therefore an advantage for the craps player to know the probability of the throwing say, a Six before a Seven, for in casino play he may back the dice either to win or lose.  Because there are five ways of throwing a Six, and six ways of throwing a Seven, the probability of throwing a Six before a Seven is 5/11.  Thus the odds are 6 to 5 against throwing a Six before a Seven.  The probability of throwing a Four (or Ten) before a Seven is 3/9 or 6 to 3 against (because there are only three ways of throwing either Four or Ten).  In the same way, for a  Five or Nine the Probability is 4/10 or 6 to 4 against; and for Eight (as for Six) it is 5/11.