
Odds are a numerical expression, always consisting of a pair of numbers, used in both gambling and statistics. In statistics, odds for reflect the likelihood that a particular event will take place. Odds against reflect the likelihood that a particular event will not take place. The uses of the term among statisticians and probabilists on the one hand, versus in the gambling world on the other hand, are not consistent with each other (with the exception of horse racing).〔〔 Conventionally, gambling odds are expressed in the form "X to Y", where X and Y are numbers, and it is implied that the odds are odds against the event on which the gambler is considering wagering. In both gambling and statistics, the 'odds' are a numerical expression of how likely some possible future event is. In gambling, odds represent the ratio between the amounts staked by parties to a wager or bet.〔 Thus, odds of 6 to 1 mean the first party (normally a bookmaker) is staking six times the amount that the second party is. Thus, gambling odds of '6 to 1' mean that there are six possible outcomes in which the event will not take place to every one where it will. In other words, the probability that X will not happen is six times the probability that it will. In statistics, the odds for an event E are defined as a simple function of the probability of that possible event E. One drawback of expressing the uncertainty of this possible event as odds for is that to regain the probability requires a calculation. The natural way to interpret odds for (without calculating anything) is as the ratio of events to nonevents in the long run. A simple example is that the (statistical) odds for rolling six with a fair die (one of a pair of dice) are 1 to 5. This is because, if one rolls the die many times, and keeps a tally of the results, one expects 1 six event for every 5 times the die does not show six. For example, if we roll the fair die 600 times, we would very much expect something in the neighborhood of 100 sixes, and 500 of the other five possible outcomes. That is a ratio of 100 to 500, or simply 1 to 5. To express the (statistical) odds against, the order of the pair is reversed. Hence the odds against rolling a six with a fair die are 5 to 1. The probability of rolling a six with a fair die is the single number 1/6 or approximately 16.7%. The gambling and statistical uses of odds are closely interlinked. If a bet is a fair one, as in a wager between friends, then the odds offered to the gamblers will perfectly reflect relative probabilities. A fair bet that a fair die will roll a six will pay the gambler $5 for a $1 wager (and return the bettor his or her wager) in the case of a six and nothing in any other case. The terms of the bet are fair, because on average, five rolls result in something other than a six, at a cost of $5, for every roll that results in a six and a net payout of $5. The profit and the expense exactly offset one another and so there is no disadvantage to gambling over the long run. If the odds being offered to the gamblers do not correspond to probability in this way then one of the parties to the bet has an advantage over the other. Casinos, for example, offer odds that place themselves at an advantage, which is how they guarantee themselves a profit and survive as businesses. The fairness of a particular gamble is more clear in a game involving relatively pure chance, such as the pingpong ball method used in state lotteries in the United States. It is much harder to judge the fairness of the odds offered in a wager on a sporting event such as a football match. ==History== The language of odds such as "ten to one" for intuitively estimated risks is found in the sixteenth century, well before the development of mathematical probability.〔 Shakespeare wrote: The sixteenth century polymath Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes〔(''Some laws and problems in classical probability and how Cardano anticipated them'' Gorrochum, P. ''Chance'' magazine 2012 )〕). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「odds」の詳細全文を読む スポンサード リンク
