Probability

Probability[pred,xdist]

gives the probability for an event that satisfies the predicate pred under the assumption that x follows the probability distribution dist.

Probability[pred,xdata]

gives the probability for an event that satisfies the predicate pred under the assumption that x follows the probability distribution given by data.

Probability[pred,{x₁,x₂,…}dist]

gives the probability that an event satisfies pred under the assumption that {x₁,x₂,…} follows the multivariate distribution dist.

Probability[pred,{x₁dist₁,x₂dist₂,…}]

gives the probability that an event satisfies pred under the assumption that x₁, x₂, … are independent and follow the distributions dist₁, dist₂, ….

Probability[pred₁pred₂,…]

gives the conditional probability of pred₁ given pred₂.

Details and Options

xdist can be entered as x dist dist or x∖[Distributed]dist.
pred₁pred₂ can be entered as pred₁ cond pred₂ or pred₁∖[Conditioned]pred₂.
For a continuous distribution dist, the probability of pred is given by ∫Boole[pred]f[x]x where f[x] is the probability density function of dist and the integral is taken over the domain of dist.
For a discrete distribution dist, the probability of pred is given by ∑Boole[pred]f[x] where f[x] is the probability density function of dist and the summation is taken over the domain of dist.
For a dataset data, the probability of pred is given by Sum[Boole[pred],{x,data}]/Length[data].
Univariate data is given as a list of values {v₁,v₂,…} and multivariate data is given as a list of vectors {{v₁₁,…,v_1m},{v₂₁,…,v_2m},…}.
Probability[pred,{x₁dist₁,x₂dist₂}] corresponds to Expectation[Probability[pred,x₂dist₂],x₁dist₁] so that the last variable is summed or integrated first.
N[Probability[…]] calls NProbability for probabilities that cannot be evaluated symbolically.
The following options can be given:

Assumptions	$Assumptions	assumptions to make about parameters
GenerateConditions	False	whether to generate conditions on parameters
Method	Automatic	what method to use

Background & Context

Probability[pred,x] represents the probability for an event that satisfies a predicate pred under the assumption that the chosen random variable x follows an indicated probability distribution (i.e.  is a discrete or continuous distribution such as NormalDistribution, BinomialDistribution, ChiSquareDistribution, etc.) or is taken from a given dataset (i.e.  defines a dataset), and where  is a shorthand for Distributed. The output of Probability is either a number (where the probability of an impossible event is 0 while the probability for a certain event is 1) or a symbolic expression involving the input parameters.
Probability also works with multivariate distributions; with nonparametric distributions such as EmpiricalDistribution, HistogramDistribution, and KernelMixtureDistribution; and with derived distributions such as TransformedDistribution and ProductDistribution. In addition, Probability may be applied to random processes, including those defined by ContinuousMarkovProcess, DiscreteMarkovProcess, WienerProcess, and PoissonProcess.
The predicate pred may include both linear and nonlinear inequalities, as well as logical combinations of inequalities. Probability can compute conditional probabilities using a predicate of the form pred₁pred₂, where  is a shorthand for Conditioned. A number of options may be passed to Probability, including Assumptions, GenerateConditions, and Method. The output provided by Probability is calculated using exact methods such as symbolic integration and summation. Corresponding results involving numerical methods can be obtained via NProbability.
Several Wolfram Language functions, including PDF, CDF, and SurvivalFunction, return results that are equivalent to those obtainable using Probability with particular predicate structures. In addition, results obtained by Probability may be obtained by Expectation by pairing Expectation with a Boole construct. For example, Expectation[Boole[1<x<3],xNormalDistribution[]] is equivalent to Probability[1<x<3,xNormalDistribution[]].
Probability can be used to exactly solve many problems in probability. For example, the probability that a toss of two dice will yield a sum of 3 or 4 is given by Probability[3≤x+y≤4,{x,y}DiscreteUniformDistribution[{{1,6},{1,6}}], while the chances that a triple of uniformly distributed random variables satisfying will yield a polynomial with real roots can be computed using Probability[b²-4a c≥0,{a,b,c}UniformDistribution[Table[{0,1},{3}]]].