NProbability

A numeric-only $P(cond)$ — handles any condition compute-engine can evaluate at a point, not just the shapes Probability recognizes symbolically.

NProbability(cond, Distributed(x, dist))tries the exact Probability first; otherwise sums (discrete) or integrates (continuous) the indicator of cond against the PDF numerically. Always returns a machine float.

Typed signatures

Type	Package	Applies
(any, expression<Distributed>) -> number	statistics	always

Domain: Statistics

Details

• Tries Probability(cond, Distributed(x, dist)) first, converted to a float, whenever it resolves.
• Otherwise sums/integrates PDF(x) over exactly the points/region where cond holds — evaluated by direct numeric substitution, so cond can be any boolean expression compute-engine can evaluate at a concrete point (Mod, Or, x^2 > 1, ...), not only Equal/Less/LessEqual/And.
• Continuous conditions are NOT generally smooth indicators (x^2 > 1 jumps at $x=\pm 1$) — plain tanh-sinh quadrature loses most of its accuracy right at a jump, so the boundary is located first (by bisection) and each constant-truth piece is integrated on its own.
• Same $10^{-12}$ tail/convergence tolerance as NExpectation; same scope gap for wave-4 compound distributions.

Examples

Basic #

$P(X^2 > 1) = 2(1-\Phi(1)) \approx 0.3173105079$ for standard normal $X$ — Probability itself stays unevaluated (x^2 > 1 isn't Equal/Less/LessEqual/And), and the indicator has two jumps ($x=\pm1$), split off before integrating. Cross-checked against wolframscript's N[NProbability[x^2 > 1, x \[Distributed] NormalDistribution[0,1]], 10].

live ↗

In$\mathrm{NProbability}(x^2\gt1, \mathrm{Distributed}(x, \operatorname{NormalDistribution}(0, 1)))$Out$0.317\,310\,507\,838\,716\,9$

Scope #

$P(X \text{ even}) = (1+e^{-2\mu})/2 \approx 0.5001677313$ for $X \sim \mathrm{Poisson}(4)$ — an infinite discrete sum, condition on Mod. Cross-checked against wolframscript's N[NProbability[Mod[x,2] == 0, x \[Distributed] PoissonDistribution[4]], 10].

live ↗

In$\mathrm{NProbability}(x\bmod2=0, \mathrm{Distributed}(x, \operatorname{PoissonDistribution}(4)))$Out$0.500\,167\,731\,313\,946\,8$

Agrees with Probability's own exact CDF-based answer for the same condition (N[92/(5 e^3)]) — the exact path is tried first and used whenever it resolves.

live ↗

In$\mathrm{NProbability}(x≤5, \mathrm{Distributed}(x, \operatorname{PoissonDistribution}(3)))$Out$0.916\,082\,057\,968\,696\,6$

Applications #

The rate reduces to $1296/125$, and the probability is a number

live ↗

In$\mathrm{NProbability}(n≤15, \mathrm{Distributed}(n, \operatorname{PoissonDistribution}(10\,368\,000×(10^{-6}))))$Out$0.937\,281\,035\,306\,783\,9$

About

Examples	4
Checked against	Wolfram Language 1 agree

See also: Probability, NExpectation, Distributed