Probability
I want to learn more about probability.
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probability is the branch of mathematics concerning events and numerical descriptions of how likely they are to occur, The probability to an event is a number between 0 and 1; the larger the probability, the more likely the event is to occur.
When dealing with random experiments - experiments that are random and well-defined - in a purely theoretical setting, probabilities can be numerically described by the number of desired outcomes, divided by the total number of all outcomes. This is referred to as theoretical probability (in contrast to empirical probability, dealing with probabilities in the context of real experiments). When it comes to practical application, there are two major competing categories of probability interpretations, whose adherents hold different views about the fundamental nature of probability:
- Objectivists
- Assign numbers to describe some objective or physical state of affairs. The most popular version of objective probability is frequentist probability, which claims that the probability of a random event denotes the relative frequency of occurrence of an experiment's outcome when the experiment is repeated indefinitely. A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome, even if it is performed only once.
- Subjectivists
- Subjectivists assign number per subjective probability. that is, as a degree of belief. The degree of belief has been interpreted as
the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E
, although that interpretation is not universally agreed upon. The most popular form of subjective probability is Bayesian probability, which includes expert knowledge as well as some experimental data to produce probabilities. The expert knowledge is represented by some (subjective) prior probability distribution. These data are incorporated in a likelihood function. The product of the prior and the likelihood, when normalized, results in a posterior probability distribution that incorporates all the information known to date.
- Subjectivists assign number per subjective probability. that is, as a degree of belief. The degree of belief has been interpreted as
The word probability derives from the Latin probabilitas
, which can also mean "probity", a measure of the authority of a witness in a legal case in Europe, and often correlated with the witness's nobility.
Like other theories, the theory of probability is a representation of its concepts in formal terms – that is, in terms that can be considered separately from their meaning.
Consider an experiment that can produce a number of results. The collection of all possible results is called the sample space of the experiment, sometimes denoted as . The power set of the sample space is formed by considering all different collections of possible results.
Independent Events
If two events, A and B are independent then the joint probability is:
Mutually Exclusive Events
If either A or event B can occur but never both simultaneously, then they are called mutually exclusive events. If two events are mutually exclusive, the the probability of both occurring is denoted and
If two events are mutually exclusive, then the probability of ether occurring is denoted and:
Not (necessarily) mutually exclusive events
If two events are not (necessarily) mutually exclusive then:
Conditional Probability
Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written and is defined by:
If , then is formally undefined by this expression.
In a deterministic universe, based on Newtonian concepts, there would be no probability if all conditions were known (Laplace's demon) (but there are situations in which sensitivity to initial conditions exceeds our ability to measure them, i.e. know them). Probability theory is required to describe quantum phenomena.
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