Poisson Calculator

Four tabs over one probability law. Single count — univariate P(X = x), its tails, and the inverse λ-solve; Joint score — shock — the common-shock bivariate P(h, a); Totals and Handicaps — the two-sided lines (quarter lines included) priced off that same score grid. Each tab is seeded to a worked example and explains its own maths inline.

↑/↓ steps an input, Shift steps bigger.

Poisson facts

P(X = x) = e^−λ · λ^x / x! · mean = λ · variance = λ · σ = √λ
Additivity: X ~ Pois(λₕ) and Y ~ Pois(λₐ) independent ⇒ X + Y ~ Pois(λₕ + λₐ) — why a match's total goals can be priced from λ_total alone.

Inputs

Average rateλ

Events expected per window — e.g. goals per match. Live σ = √λ = 1.6353.

0 < λ ≤ 99

Countx

The number of events to evaluate at.

whole number, 0 … 120

Probabilities at x

Quantity	Meaning	Value	%	Fair dec
P(X = 3)	exactly 3 events	0.21981	21.98%	4.549
P(X < 3)	2 or fewer	0.49994	49.99%	2.000
P(X ≤ 3)	3 or fewer	0.71975	71.97%	1.389
P(X > 3)	4 or more	0.28025	28.03%	3.568
P(X ≥ 3)	3 or more	0.50006	50.01%	2.000

How P(X = x) is computed

0	start at zero events: P(X = 0) = e^−λe^−2.6743 — the chance the window stays empty	0.06896
1	climb to 1: multiply by λ/1 = 2.6743P(X = 1) = P(X = 0) × λ/1	0.18441
2	climb to 2: multiply by λ/2 = 1.3372P(X = 2) = P(X = 1) × λ/2	0.24658
3	climb to 3: multiply by λ/3 = 0.8914P(X = 3) = P(X = 2) × λ/3	0.21981
✓	direct formula check: e^−λ · λ^x / x!0.068955 × 19.126274 ÷ 6	0.21981

Solve for λ — supply a probability, get the rate back

TailP(X ⊙ x)

Which probability statement the target refers to. P(X ≥ x) is the over-line case: P(over x − 0.5).

Countx

The count in the probability statement.

whole number, 0 … 120

Targetp

The probability the statement should have.

0 < p < 1 — type 0.5 or 50%

Solved λ	the rate at which "3 or more events" happens 50.00% of the time	2.6741
σ = √λ	spread of the count at that rate	1.6353
Check	P(X ≥ 3) at λ = 2.6741	0.50000 = the target ✓

How λ is solved — a higher-or-lower search

·	no formula maps a target probability back to λ — but P(X ≥ 3) only rises as λ grows, so guessing is safeguess a rate, compare P(X ≥ 3) with the 50.00% target, keep the half of the window that must hold the answer
1	window 0.00 – 99.00 → guess the midpoint, λ = 49.5001P(X ≥ 3) at this rate is 100.00% — above the 50.00% target; this tail rises as λ grows, so the answer is lower	▼ keep lower half
2	window 0.00 – 49.50 → guess the midpoint, λ = 24.7501P(X ≥ 3) at this rate is 100.00% — above the 50.00% target; this tail rises as λ grows, so the answer is lower	▼ keep lower half
3	window 0.00 – 24.75 → guess the midpoint, λ = 12.3751P(X ≥ 3) at this rate is 99.96% — above the 50.00% target; this tail rises as λ grows, so the answer is lower	▼ keep lower half
4	window 0.00 – 12.38 → guess the midpoint, λ = 6.1876P(X ≥ 3) at this rate is 94.59% — above the 50.00% target; this tail rises as λ grows, so the answer is lower	▼ keep lower half
5	window 0.00 – 6.19 → guess the midpoint, λ = 3.0938P(X ≥ 3) at this rate is 59.75% — above the 50.00% target; this tail rises as λ grows, so the answer is lower	▼ keep lower half
6	window 0.00 – 3.09 → guess the midpoint, λ = 1.5470P(X ≥ 3) at this rate is 20.30% — below the 50.00% target; this tail rises as λ grows, so the answer is higher	▲ keep upper half
…	and so on — each guess halves the window; 60 halvings pin λ beyond display precision	…
✓	converged λcheck: P(X ≥ 3) = 0.50000 = the target	2.6741

Range probability

Minimumx₁

Lower bound of the inclusive range.

whole number, 0 … 120

Maximumx₂

Upper bound of the inclusive range.

whole number, x₁ … 120

Quantity	Meaning	Value	%	Fair dec
P(2 ≤ X ≤ 3)	inside the range, ends included	0.46639	46.64%	2.144
P(X < 2) + P(X > 3)	outside the range — both tails	0.53361	53.36%	1.874

Distribution table

x	P(X = x)	P(X ≤ x)
0	0.06896	0.06896
1	0.18441	0.25336
2	0.24658	0.49994
3	0.21981	0.71975
4	0.14696	0.86671
5	0.07860	0.94531
6	0.03503	0.98035
7	0.01338	0.99373
8	0.00447	0.99820
9	0.00133	0.99953
10	0.00036	0.99989
11	0.00009	0.99998
> 11	Σ 0.00002	1.00000