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Statistics

 Probability
 compute the probability of an event

 Calling Sequence Probability(X, options)

Parameters

 X - algebraic, relation, or set of algebraics and relations, each involving at least one random variable; an event options - (optional) equation of the form numeric=value; specifies options for computing the probability density function of a random variable

Description

 • The Probability command computes the probability of the event X.
 • The first parameter, X, is an event consisting of a relation or set of relations. An algebraic expression is interpreted as an equation set to zero. Each relation must involve at least one random variable. All random variables in X are considered independent. A set is interpreted as the intersection of the events of each of its members.

Computation

 • By default, all computations involving random variables are performed symbolically (see option numeric below).

Options

 The options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.
 • numeric=truefalse -- By default, the probability density function is computed using exact arithmetic. To compute the probability density function numerically, specify the numeric or numeric = true option.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$

Compute the probability of the normal distribution.

 > $X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(0,1\right)\right):$
 > $\mathrm{Probability}\left({X}^{2}<1\right)$
 ${\mathrm{erf}}{}\left(\frac{\sqrt{{2}}}{{2}}\right)$ (1)
 > $\mathrm{Probability}\left({X}^{2}<1,'\mathrm{numeric}'\right)$
 ${0.682689492137086}$ (2)

Compute the probability that the product of 3 independent random variables uniformly distributed on between 0 and 1 is less than t.

 > $X≔\left[\mathrm{seq}\left(\mathrm{RandomVariable}\left(\mathrm{Uniform}\left(0,1\right)\right),i=1..4\right)\right]:$
 > $Y≔X\left[1\right]X\left[2\right]X\left[3\right]:$
 > $\mathrm{Probability}\left(Y
 $\left\{\begin{array}{cc}{0}& {t}{\le }{0}\\ \frac{{{\mathrm{ln}}{}\left({t}\right)}^{{2}}{}{t}}{{2}}{-}{t}{}{\mathrm{ln}}{}\left({t}\right){+}{t}& {t}{\le }{1}\\ {1}& {1}{<}{t}\end{array}\right\$ (3)

Compute the probability that the distance between two points randomly chosen from a 1x1 square is less than 1.

 > $Z≔{\left({\left(X\left[1\right]-X\left[3\right]\right)}^{2}+{\left(X\left[2\right]-X\left[4\right]\right)}^{2}\right)}^{\frac{1}{2}}:$
 > $\mathrm{Probability}\left(Z<\frac{1}{2}\right)$
 ${-}\frac{{29}}{{96}}{+}\frac{{\mathrm{\pi }}}{{4}}$ (4)

References

 Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.