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In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. [1] A single outcome may be an element of many different events, [2] and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. [3]
The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the belief that, if an event (whose occurrences are independent and identically distributed) has occurred less frequently than expected, it is more likely to happen again in the future (or vice versa).
Individual random events are, by definition, unpredictable, but if there is a known probability distribution, the frequency of different outcomes over repeated events (or "trials") is predictable. [note 1] For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often ...
The probability of an event is then defined to be the sum of the probabilities of all outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is (“ ”) + (“ ”) + (“ ”) = + + = .
For example, when tossing an ordinary coin, one typically assumes that the outcomes "head" and "tail" are equally likely to occur. An implicit assumption that all outcomes are equally likely underpins most randomization tools used in common games of chance (e.g. rolling dice , shuffling cards , spinning tops or wheels, drawing lots , etc.).
Usually, coincidences are chance events with underestimated probability. [3] An example is the birthday problem, which shows that the probability of two persons having the same birthday already exceeds 50% in a group of only 23 persons. [4] Generalizations of the birthday problem are a key tool used for mathematically modelling coincidences. [5]
A probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) is assigned a value of one. To qualify as a probability, the assignment of values must satisfy the requirement that for any collection of mutually exclusive ...
The mathematics of gambling is a collection of probability applications encountered in games of chance and can get included in game theory.From a mathematical point of view, the games of chance are experiments generating various types of aleatory events, and it is possible to calculate by using the properties of probability on a finite space of possibilities.