Carddan’s historic achievement was to crystallize a rule: Events may be graded into three kinds of card game : (a) the “impossible” (such as throwing a Seven with a single die); (b) “certain” (such as the fact that one side of a thrown die must fall uppermost); (c) the “probable”(such as a Six falling uppermost on the first throw of a die).  If (a) is expressed arithmetically as 0 and (b)is expressed as 1, then all the degrees of probability in between can be expressed as fractions.

During the running of the 1962 British Derby, the “favorite” was brought down in a pile-up involving six horses.  The odds against the eventual poker winner were 22 to 1.  Such unpredictable accidents show why Cardan’s probability formula can be applied only to games of pure chance (like coin tossing or roulette) in which each possible case has an equal chance of winning.

The important thing to remember about this elementary rule is that it is applicable only in games of pure chance (like dice, roulette, or lotteries).  It would be foolish to forget Cardan’s phrase “equally likely cases” and attempt to apply the rule to, say, a race between six horses.  One horse may be faster than another, or better ridden, or have more endurance, or may fall down, or turn and run the wrong way.  Many factors (for instance, human and animal temperament, terrain, weather, or weight) can affect the result.  The only reasonable certainties after the “off” are that six horses are running and that one (or possibly two)will get to the winning post before the others.
So to bet on the result of a race you must choose one of two courses:  Either you can muster up all your knowledge of these particular horses and jockeys and all the relevant factors (previous performance, ability of jockey, character of the track, odds offered, and so on); or you can place you bet blindly making your choice for some arbitrary reason (a “hunch” or the color of the jockey’s shirt).  It is of course true that probabilities are worked out arithmetically and offered as odds on or against a particular horse’s winning; but they are calculated by the bookmakers from experience and in accordance with the amount of money that is bet (see Chapter 8), not from the p = f/c law.

A simplified diagram of the “law of large lucky numbers ” in terms of coin tossing.  The difference between the number of heads and tails thrown tends to decrease as the number of tosses increases.  When the number of tosses nears 100,000,000, the totals of heads and tails thrown may be expected to approach equality.  With an infinite number of tosses, the totals (in theory ) would even up.

One more observation is necessary about Cardan’s basic law of probability.  It concern a very important characteristic of the law-and  one that is very often forgotten or ignored by most people (who instead wryly substitute the altogether different element of “luck”): In a game of chance (where all the possible cases are equally likely and all the favorable cases are known, as in dice, coin tossing, or roulette) it may seem reasonable to assume that all the players have an equal chance of winning.  And, according to the probability law, they have-in the long run.

“The long run is a popular phrase used as a synonym for a general mathematical law called the Law of Large Number.  To illustrate this law in simple terms: In, say, 10,000 tosses of a coin the proportion of heads will deviate by less than 1 per cent from the probable proportion of 1/2 .  As the number of tosses increases, the deviation from equal proportions of heads and tails decreases.
That is, the proportions theoretically tend to even up as the number of tosses approaches infinity.  So the long run isn’t simply a stretch of coin tossing with a beginning and an end.  It is an interminability of coin tossing, from an unspecified first toss somewhere in the past to never-to-be-achieved last one in the future.
To understand the long run in relation to dice, imagine a circle representing the totality of dice games past, present, and future-with all the players standing on the periphery throwing dice.  If it were possible for all those players to stay permanently on that circle and throw that infinite number of throws, they would all have precisely equal chances of winning, just as the six equal sides of their dice would each fall uppermost an equal number of times.  But since players constantly join, leave, and rejoin the circle, and since the number of games remains indeterminate (because the future hasn’t happened yet), each player adding to this infinity of throws has to accept the specific variation in chance that happens to be current as he joins.
The variation may be working against him so far as financial success is concerned.  But if you ignore all the stakes laid in all the games around the circle, you will see that “for” and “against” are meaningless terms where chance (a natural law) is concerned.  Chance is heedless of artificial contrivances like dice or money or evaluations of “for” and “against.”  Chance brings about natural events like the distribution of geniuses among earthquake victims, or the incidence of black cats in gamblers’ paths.  It is man that decides whether these events are valuable (“for”) or harmful (“against”).
A player in a bridge game who is dealt a hand consisting of 13 Spades the complete suit may think the event remarkable.  But the odds against his being dealt a complete run of a single suit are exactly the same as the odds against his being dealt any other specified combination of 13 cards (specifically, 635,013,559,599 to 1).  And the card player who talks of “a run of bad luck” (when he picks up several consecutive hands of cards that are inconveniently useless as winning hands according to the rules of the poker game) must remember that chance is as heedless of the rules of bridge as of the existence of money.  His chances of picking up winning hands are exactly the same as the other player’s chances in the long run; but not necessarily in the same evening games.  It follows, then, that the longer he plays the better are his chances of evening up with his opponents in the drawing of winning hands.
Although the true long run is an infinite number of games, it is clearly impracticable to consider it as such.  In practice a run can be considered only relatively long.  But the principle remains valid.  Professional gamblers who act as bookmakers or run casinos are in the position of playing against more or less unlimited time and wealth (represented by continual bets placed by innumerable people).  They are therefore  forced to give themselves an artificial advantage, either by charging a fee for each bet they accept, or by paying off their losses at less than the proper odds.  I’ll explain the workings of that advantage later.  At the moment I want to urge you to remember the importance of the long run.

After Cardan, many great minds applied their mathematical powers to the task of extending the theories of chance and probability, among them the 17th century French mathematician and thinker Blaise pascal.  Several widely varying claims to fame are allowable to Pascal: his Pensées; the differential calculus (which he cleared the way to); the early blooming of his mathematical genius (his sister said that as a mere child he rediscovered the 32 theorems of geometry); the calculating machine (which he invented); his religious fervor; and his solutions to the gambling problems of a gamester called Antoine Gombaud, the Chealier de méré had rightly deduced, or perhaps guessed, that the odds favored his throwing a Six at least once in four throws of a die.  He had won himself a lot of money on that proposition.  Multiplying chances, he had wrongly deduced that with two dice the odds would favor his throwing at least one double Six in every 24 throws.  Unaccountably this idea has worked against him, and he was going bankrupt.  So he turned to Pascal for help.