100 gambling edition

Chances, probabilities, and odds Events or outcomes that are equally probable have an equal chance of occurring in each instance. In games of pure chance, each instance is a completely independent one; that is, each play has the same probability as each of the others of producing a given outcome. Probability statements apply in practice to a long series of events but not to individual ones. The law of large numbers is an expression of the fact that the ratios predicted by probability statements are increasingly accurate as the number of events increases, but the absolute number of outcomes of a particular type departs from expectation with increasing frequency as the number of repetitions increases. It is the ratios that are accurately predictable, not the individual events or precise totals. The probability of a favourable outcome among all possibilities can be expressed: probability (p) equals the total number of favourable outcomes (f) divided by the total number of possibilities (t), or p = f/t. But this holds only in situations governed by chance alone. In a game of tossing two dice, for example, the total number of possible outcomes is 36 (each of six sides of one die combined with each of six sides of the other), and the number of ways to make, say, a seven is six (made by throwing 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, or 6 and 1); therefore, the probability of throwing a seven is 6/36, or 1/6. In most gambling games it is customary to express the idea of probability in terms of odds against winning. This is simply the ratio of the unfavourable possibilities to the favourable ones. Because the probability of throwing a seven is 1/6, on average one throw in six would be favourable and five would not; the odds against throwing a seven are therefore 5 to 1. The probability of getting heads in a toss of a coin is 1/2; the odds are 1 to 1, called even. Care must be used in interpreting the phrase on average, which applies most accurately to a large number of cases and is not useful in individual instances. A common gamblers’ fallacy, called the doctrine of the maturity of the chances (or the Monte-Carlo fallacy), falsely assumes that each play in a game of chance is dependent on the others and that a series of outcomes of one sort should be balanced in the short run by the other possibilities. A number of systems have been invented by gamblers largely on the basis of this fallacy; casino operators are happy to encourage the use of such systems and to exploit any gambler’s neglect of the strict rules of probability and independent plays. An interesting example of a game where each play is dependent on previous plays, however, is blackjack, where cards already dealt from the dealing shoe affect the composition of the remaining cards; for example, if all of the aces (worth 1 or 11 points) have been dealt, it is no longer possible to achieve a “natural” (a 21 with two cards). This fact forms the basis for some systems where it is possible to overcome the house advantage.
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