Conditional probability

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This article defines some terms which characterize probability distributions of two or more variables.

Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the probability of A, given B".

Joint probability is the probability of two events in conjunction. That is, it is the probability of both events together. The joint probability of A and B is written [P(A \cap B)] or [P(A, B).]

Marginal probability is the probability of one event, regardless of the other event. Marginal probability is obtained by summing (or integrating, more generally) the joint probability over the unrequired event. This is called marginalization. The marginal probability of A is written P(A), and the marginal probability of B is written P(B).

In these definitions, note that there need not be a causal or temporal relation between A and B. A may precede B, or vice versa, or they may happen at the same time. A may cause B, or vice versa, or they may have no causal relation at all.

Conditioning of probabilities, i.e. updating them to take account of (possibly new) information, may be achieved through Bayes' theorem.

Definition

If A and B are events, and P(B) > 0, then the conditional probability of A, given B is

[P(A \mid B)= \frac.]

Equivalently, we have

[P(A \cap B)=P(A\mid B) P(B).]

Statistical independence

Two random events A and B are statistically independent if and only if

[P(A \cap B) \ = \ P(A) P(B).]

Thus, if A and B are independent, then their joint probability can be expressed as a simple product of their individual probabilities.

Equivalently, for two independent events A and B,

[P(A|B) \ = \ P(A)]

and

[P(B|A) \ = \ P(B).]

In other words, if A and B are independent, then the conditional probability of A, given B is simply the individual probability of A alone; likewise, the probability of B given A is simply the probability of B alone.

Mutual exclusivity

Two events A and B are mutually exclusive if and only if

[P(A \cap B) = 0]

as long as

[P(A) \ne 0]

and

[P(B) \ne 0.]

Then

[P(A\mid B) = 0]

and

[P(B\mid A) = 0.]

In other words, the probability of A happening, given that B happens, is nil since A and B cannot both happen in the same situation; likewise, the probability of B happening, given that A happens, is also nil.

Other considerations

• If [B] is an event and [P(B) > 0], then the function [Q] defined by [Q(A) = P(A|B)] for all events [A] is a probability measure.

• If [P(B)=0], then [P(A|B)] is left undefined.

• Conditional probability can be calculated with a decision tree.

See also

• Bayes' theorem
• Likelihood function
• Posterior probability
• Probability theory
• Monty Hall problem

	Probability distributions  [ view] • [ talk] • [ edit]
	Univariate	Multivariate
Discrete:	Bernoulli • binomial • Boltzmann • compound Poisson • degenerate • degree • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial
Continuous:	Beta • Beta prime • Cauchy • chi-square • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square • inverse gaussian • inverse gamma • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Voigt • von Mises • Weibull • Wigner semicircle	Dirichlet • Kent • matrix normal • multivariate normal • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling

 

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