The soldier in a shellhole who thinks "I'm safe here because mathematics has eliminated the possibility of another shell's falling on the same spot's is perpetuating exactly the same fallacy as the casual gambler (call him Lucky Jim) at a roulette table who observes that red has come up for 10 consecutive spins, and who therefore puts his bet on black because he thinks it simple must come up on the 11th spin.  The probability of black has not been increased in the slightest by the 10 preceding reds; for (this is a basic poker rule in any game of pure chance) every single cast of a die, toss of a coin, or spin of a roulette wheel is completely unrelated to every other cast, toss, or spin, whether before or after.  In other words, the chances of Jim's winning on any particular spin of the wheel are not improved in the slightest by his knowledge of the results of the 10 (or 100, or 1000) previous spins.
At this point I had better deal with a possible objection to that statement "Surely," some Lucky Jim might say, "backgammon ought to come up on the 11th spin or soon afterward by the law of averages!" But the phrase "law of averages"  is often wrongly used, and in this context is meaningless.  What most people mean by the "law of averages" is the "law of large numbers," which (as we have seen) says that in the long run all the possible cases will happen an equal number of times.  In terms of roulette, the law would say that, as the number of spins of the wheel approaches infinity, the number of blacks tends to even up with the number of reds.

Napoleon maintained that he could mathematically eliminate chance from a battle by carefully calculating every move and countermove beforehand.  But events proved that chance can't be eliminated.  Left, a battle plan of Waterloo.

But 11 spins of the wheel do not constitute a long run.  So Lucky Jim, play poker betting on black on the 11th spin, has no need to concern himself with the long run, or the law of large numbers, or with any mathematical law other than this one:  The probability of black's coming up on any one spin remains the same no matter what were the results of preceding spins-a probability of ?.

Lucky Jim's are usually much more reasonable about their expectations in lottery gambles.  Lotteries (and other forms of numbers gamble), though offering a prize distribution that is dependent on pure chance, are not participating games from Jim's point of view.  He merely buys a ticket, knowing that the single event of the drawing of the lottery will decide whether he wins or not.  If there were 100 different lotteries, all being drawn at consecutive moments on the same day, and if he had a ticket in each of them, he would not expect the numbers of the first 00 tickets drawn to affect the number of the 100th ticket.  Yet that is what he is expecting of the random whims of chance if he supposes that 10 consecutive reds in roulette can influence the result of the 115th spin of the wheel.
(One noteworthy fact about lotteries and other numbers games, though:  Each time a ticket is drawn in a multiple-prize lottery or pool, the odds against the success of the remaining numbers are reduced.  If there are a million tickets in the lottery, the odds against a specific number's being drawn first time are 999,999 to 1, and they are reduced by one each time a ticket is drawn.  But in a million-ticket lottery this is scarcely an encouraging reduction in odds.)

It must not be assumed that the few basic elements of the theory of probability that I've discussed so far- odds, favorable percentages, the long run, etc. - are all there is to it.  Theory had grown into quite a jungle of complicated mathematics even by Pascal's day; and, after him, extensions and revisions to the theory were made by, among others, Huygens, Jacques and Daniel Bernoulli, Newton, d'Alembert, and Gauss, while the French astronomer Pierre Laplace co-ordinated the findings of more than a dozen mathematicians over two centuries in his Theorie Analytique des Probabiliies , and included his own application of the theory to the causes of phenomena and statistics.
Huygens, a 17th century Dutchman, was a an early statistician who paved the way for life insurance by working out tables relating to the probable expectation of life at various ages.  The bernoullis-a Swiss family of mathematicians who spread their achievements over the 17th and 18th centuries -polularized the newly discovered calculus of Newton and Leibnitz, worked out innumerable calculations concerning hydraulics and pneumatics, and systematically studies every known field of mathematics, including one very important to the poker theory of probability:  the field of large numbers.  This particular exploration has come to be known as the "Bernoullian Series of Bernoullian Numbers"; it plays a part in my brief look at statistics later.  The German mathematician Gauss produced the Law of Errors, which stated that in repeated measurement of the same object or process, it must not be expected that the outcome of every observation will be the same.  Newton's revelation of the differential calculus and d'Alembert's studies of the laws of dynamics both opened new possibilities for the development of the theory of probability; but these discoveries get far too complicated in their mathematics for summarization here.

All these towering scientists had been concerned with probability in the line of their studies as physicists, astronomers, geodesists, and mathematicians.  But their work opened the way for the famous gambler, and also for the insurers, bankers, advertising men, nuclear scientists, and doctors of today.  The theory of probability is used in determining the behavior of atoms, in working out problems of genetics, in fixing life-insurance rates based on tables of mortality, in conducting opinion polls, in advertising detergents to the likeliest buyers, in establishing the life cycles of bacteria and political parties, and in predicting the success of all kinds of commercial enterprises.  But it is used, in some of these cases, in  a rather different way- i.e., in association with the science of statistics.
The observation that statistics can be made to prove anything is a euphemistic version of Disraeli's remark: "There are lies, damned lies, and statistics."  As a generalization it isn't true, though it is true that a statistician can, if he wishes, forget or suppress important factors in statistical analyses and thereby create false impressions.  However, giving such rouges no more than that brief mention, I'll turn now to the proper uses of the science (simplified to avoid the complex and esoteric mathematics that are involved)

Statistics are measurable facts recorded in figures and interpreted with some special object in mind.  The objects may be extremely varied.  I have mentioned opinion polls, insurance, advertising, medicine, physics, politics, and commerce as activities depending on, or at any rate using, statistics. But there are other more surprising uses for them.  For example, they have several times been used to establish the disputed authorship of literary works.