A Chart for the blackjack online poker player showing the number of ways (with two cards) of making each number between 13 and 21.  (In blackjack, a player tries to obtain a higher total card count than the dealer without exceeding a total of 21)

Even these runs of luck are predicted by the laws of probability.  They manifest themselves (so far as life insurance is concerned ) in unusual numbers of deaths due to such occurrences as war, famine, and pestilence.  These are deviations from expectation.  Similar deviation can be predicted for any event governed by chance.  The American mathematician Arne Fisher, in his The mathematical Theory of Probabilities, describes an experiment in which 10,000 drawings were made from a pack of cards.  Since there are an equal number of red and black cards, the expectation was that each color would be drawn 5000 times.  In fact 4933 blackcards were drawn –a deviation of 67.  The deviation predicted for 10,000 cases by Gauss’s Law of Errors was 33.7,  so apparently something went adrift with the experiment.  But the Bernoullian Numbers (Jacques Bernoulli’s series of numbers that determine the frequency distribution of events) must also be used when calculating predictable errors; and the combination of these two laws, applied to the result of the experiment, proved that a deviation of about twice the normal is expected once in five and a half times.

All the possible applications of all the laws of probability cannot, of course, give reliable figures to insurance companies unless the mortality tables to which they are applied are themselves reliable.  James Dodson, an 18th century English pioneer actuary, worked out a table premiums payable annually to ensure $ 100 on death at any age between 14 and 67; but the mortality tables of those days were so inaccurate that the premiums were fixed at too high a rate and Dodson’s company accumulated huge reserves of money.  (It was later returned in the form of bonuses.)
This situation would be unlikely to arise today.  Even if there were no laws regulating profits, an insurance company whose premiums were too high would simply lose customers, as would a casino whose “favorable percentage” was too high.  So the premium or the percentage must be of a size that will in the long run give the company or casino a profit but won’t alienate potential customers.  In gambling casinos, however, the players are not totally at the mercy of the house’s favorable percentage.  Many gamblers try to put the laws of probability to work on their side of the card table or european roulette wheel by devising “systems” that are supposed to increase their chances of winning .  As will be seen in later chapters, there are several things that can make the most cunningly devised system go awry.  Not the least of these is the excitement of play itself, which can make a gambler throw his mathematics to the winds and plunge.
But some gambling systems have proved to be reasonably reliable.  The ordinary player of the card game called blackjack (or twenty-one)  may not be greatly helped by the basic knowledge that there are 1326 different ways in which two cards can total the numbers from two to 21 and that there are only 564 two-card combinations worth 16 or more. but he can do better than that, if his mathematics and memory are up to it, by using a system developed by Edward ). Thorp, an American mathematics professor.

A chart for four three-card monte hands(which total- left to right- 16, 15, 14, and 13) showing the chances of each hands reaching a count of between 17 and 21 with any number of extra cards; exceeding 21 with any number of extra cards; and exceeding 21 with one extra card.

Blackjack is a relatively simple game: Basically, the player must take cards from the dealer (or banker) that total 21 or less, and he wins if his total is higher than the dealer’s.  The ace counts one or 11; court cards count 10; other cards count their face value.  In a two-handed game this is the usual procedure: The dealer gives his opponent and himself one card each, face down.  Then he gives his opponent a card face up.  The opponent can “stay” (i.e., refuse additional cards) or he can ask for more cards to increase his total.  (If his total goes over 21 he loses automatically.)  When the opponent stays, hemakes his bet.  Then the dealer gives himself a card or cards (face up), depending on what he needs to increase his total (but to stay under 21), and covers the bet.

Thorp based his system on the fact that the cards usually aren’t shuffled after each hand of blackjack card game . Thus when certain combinations of cards have been used, the odds (or the “favorable percentages”) favor the player rather than the dealer.  For instance, when the fives are gone, the player has a favorable percentage of 3.3 per cent; but the dealer has an edge of 2.7 per cent when the aces are gone.
The player must keep track of all cards played; he must also be aware of the changes in advantage when different combinations of cards are gone.  For this knowledge the player would have to follow Thorp’s example and use a computer to work out all the various fluctuations possible, and the size of the percentage in each case.  But armed with these facts, a player would be able to know whether or not he had an edge after the first hand, and could judge his cards and bet accordingly.
It is doubtful if a more complex system has even been devised.  But Thorp’s own experiences (he has promised a book on the subject) seem to indicate that few other systems have been so successful.  Even when betting cautiously, Thorp was able to win $2000 in less than four hours in a Las Vegas club.  And his success was capped when he was barred from several casinos in the same city – because he was winning too steadily.  Most other gamblers’ systems seem excellent in theory but prove imperfect in practice.  Which perhaps is just as well or all the excitement of gambling might be lost in a flood of mathematics.