The Probability function 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.

If the option inert is not included or is specified to be inert=false, then the function will return the actual value of the result. If inert or inert=true is specified, then the function will return the formula of evaluating the actual value.

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

If there are floating point values or the option numeric is included, then the computation is done in floating point. Otherwise the computation is exact.

By default, the probability of an event is computed according to the rules mentioned above. To always compute the probability numerically, specify the numeric or numeric = true option.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Statistics}\right]\right)\:$

$N≔\mathrm{NormalRandomVariable}\left(a\,b\right)\:$

$\mathrm{Probability}\left(x<N\right)$

$\frac{1}{2}-\frac{\mathrm{erf}\left(\frac{\left(x-a\right)\sqrt{2}}{2b}\right)}{2}$

$\mathrm{Probability}\left(\mathrm{NormalRandomVariable}\left(2\&comma;3\right)<x\right)$

$\frac{1}{2}+\frac{\mathrm{erf}\left(\frac{\left(x-2\right)\sqrt{2}}{6}\right)}{2}$

$\mathrm{Probability}\left(\mathrm{NormalRandomVariable}\left(2\&comma;3\right)<1\right)$

$\frac{1}{2}-\frac{\mathrm{erf}\left(\frac{\sqrt{2}}{6}\right)}{2}$

$\mathrm{Probability}\left(\mathrm{NormalRandomVariable}\left(2\&comma;3\right)<1\&comma;'\mathrm{numeric}'\right)$

$0.369441340181764$

$E≔\mathrm{ExponentialRandomVariable}\left(5\right)\&colon;$

$\mathrm{Probability}\left(\left\{4<E\&comma;E<7\right\}\right)$

$-{\&ExponentialE;}^{-\frac{7}{5}}+{\&ExponentialE;}^{-\frac{4}{5}}$

$X≔2\mathrm{BernoulliRandomVariable}\left(\frac{1}{2}\right)+\mathrm{ExponentialRandomVariable}\left(3\right)\&colon;$

$\mathrm{Probability}\left(X<5.0\right)$

$0.7216224780$

$\mathrm{Probability}\left(X<5\&comma;\mathrm{inert}\right)$

$\underset{u\to 5-}{lim}\sum _{\mathrm{\_k}=0}^1\left(\left\{\begin{array}{cc}0& \mathrm{\_k}<0\\ 0& 1<\mathrm{\_k}\\ \frac{1}{2}& \mathrm{otherwise}\end{array}\right.\right)\left({\int }_{-\mathrm{\infty }}^{u-2\mathrm{\_k}}\left(\left\{\begin{array}{cc}0& \mathrm{\_t}<0\\ \frac{{\&ExponentialE;}^{-\frac{\mathrm{\_t}}{3}}}{3}& \mathrm{otherwise}\end{array}\right.\right)\&DifferentialD;\mathrm{\_t}\right)$

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

The Student[Statistics][Probability] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.