Keep the binomial upper tail precise instead of rounding it to zero

Essentials.DistributionProviders.Binomial/BinomialDistributionProvider.cs

@@ -1,4 +1,4 @@
-// Copyright (c) 2023-2026 ktsu-dev contributors
+// Copyright (c) 2023-2026 ktsu-dev contributors
 
 namespace ktsu.Essentials.DistributionProviders.Binomial;
 
@@ -118,6 +118,21 @@ public double Cdf(int value)
 			: SpecialFunctions.RegularizedIncompleteBeta(Trials - value, value + 1.0, failureProbability);
 	}
 
+	/// <inheritdoc />
+	public double SurvivalFunction(int value)
+	{
+		if (value < 0)
+		{
+			return 1.0;
+		}
+
+		// The upper tail is P(X > k) = I(p; k + 1, n - k). Taking it as one minus the CDF would round
+		// every tail below about 1e-16 to zero, and those small tails are usually the reason to ask.
+		return value >= Trials
+			? 0.0
+			: SpecialFunctions.RegularizedIncompleteBeta(value + 1.0, Trials - value, Probability);
+	}
+
 	/// <inheritdoc />
 	public int Sample(IRandomProvider random)
 	{

Essentials.Tests/DistributionProviderTests.cs

@@ -1,4 +1,4 @@
-// Copyright (c) 2023-2026 ktsu-dev contributors
+// Copyright (c) 2023-2026 ktsu-dev contributors
 
 namespace ktsu.Essentials.Tests;
 
@@ -622,6 +622,23 @@ public void Binomial_Stays_Finite_Where_The_Coefficient_Would_Overflow()
 		Assert.AreEqual(1.0, (2.0 * distribution.Cdf(999)) + distribution.Pmf(1000), 1e-11);
 	}
 
+	[TestMethod]
+	public void Binomial_Keeps_Its_Precision_Far_Out_In_The_Upper_Tail()
+	{
+		// Both values were computed exactly, as rationals, outside this codebase. One minus the CDF rounds
+		// the first to zero and keeps only eight digits of the second.
+		IDiscreteDistribution distribution = new BinomialDistributionProvider(100, 0.01);
+		Assert.AreEqual(9.576655593219767e-22, distribution.SurvivalFunction(20), 9.576655593219767e-22 * 1e-12);
+		Assert.AreEqual(6.255518382834089e-09, distribution.SurvivalFunction(10), 6.255518382834089e-09 * 1e-12);
+
+		Assert.AreEqual(1.0, distribution.SurvivalFunction(-1), Tight);
+		Assert.AreEqual(0.0, distribution.SurvivalFunction(100), Tight);
+		IDiscreteDistribution never = new BinomialDistributionProvider(5, 0.0);
+		Assert.AreEqual(0.0, never.SurvivalFunction(0), Tight);
+		IDiscreteDistribution always = new BinomialDistributionProvider(5, 1.0);
+		Assert.AreEqual(1.0, always.SurvivalFunction(4), Tight);
+	}
+
 	[TestMethod]
 	public void Poisson_Matches_Values_Summed_Independently()
 	{