Getting it into your agent
One page per mod, every tool's command on it. A separate URL per tool would split the same page into five that compete with each other.
npx skills add hamzabellouch/agent-skills --skill probability-theorygit clone --depth 1 https://github.com/hamzabellouch/agent-skillsWrote this? Show the measurements
A badge with what this costs and how it scanned, read live from this page, so it follows the numbers instead of freezing them. Markdown for a README, HTML for a documentation site or a project page.
[](https://agentmods.dev/skills/hamzabellouch/agent-skills/probability-theory)<a href="https://agentmods.dev/skills/hamzabellouch/agent-skills/probability-theory"><img src="https://agentmods.dev/badge/skills/hamzabellouch/agent-skills/probability-theory/github.svg" alt="Measured on agentmods" height="20"></a>Or the 80×15 button, for a site that already has a row of RSS and ATOM ones. Only the verdict fits; the numbers stay here.
<a href="https://agentmods.dev/skills/hamzabellouch/agent-skills/probability-theory"><img src="https://agentmods.dev/badge/skills/hamzabellouch/agent-skills/probability-theory.svg" alt="Reviewed on agentmods" width="80" height="20"></a>What it costs to keep this loaded
Counted locally with the o200k_base tokenizer, which is exact for GPT models; Claude uses its own tokenizer and its counts differ. Treat this as one consistent yardstick across the catalogue rather than a bill. Prices are per million input tokens.
| Model | Per session | Once invoked |
|---|---|---|
| Fable 5.1 | $0.00062 | $0.00728 |
| Opus 5.5 | $0.00025 | $0.00291 |
| Sonnet 5.5 | $0.00012 | $0.00146 |
| Haiku 4.5 | $0.00006 | $0.00073 |
Grade A, and why
probability-theory scanned grade A with 0 findings against 26 rules in 11 categories — prompt injection, anti-refusal, data exfiltration, privilege escalation, supply chain, agent snooping, system-prompt leakage, SSRF and excessive agency — measured today.
A static scan of the body, not an audit. Every finding is printed with the line that produced it so you can judge whether it matters here. A mod is markdown that instructs an agent; that is exactly why what it instructs is worth reading.
Nothing flagged
None of the 26 patterns this scan looks for appear in this file: no shell pipes, no recursive deletes, no credential paths, no hidden text, no instruction-override or anti-refusal phrasing, no agent-config snooping. That is not a guarantee, it is the absence of the things that are checkable.
How it starts
The opening of the file, as written. The whole thing — 33 lines — stays where its author put it; the contents beside it link to each section on GitHub.
Probability Theory
Foundations
- Axioms (Kolmogorov): P(Omega) = 1, P >= 0, countable additivity. Inclusion-exclusion; complement.
- Conditional probability P(A|B) = P(A and B)/P(B); independence; Bayes P(A|B) = P(B|A)P(A)/P(B); total probability. Beware base-rate neglect, Monty Hall (switching gives 2/3), Simpson's paradox, birthday paradox (23 people ~ 50.7%).
- Counting: permutations n!, combinations C(n,k), multinomial, stars and bars C(n+k-1, k-1), derangements n! sum (-1)^k/k!.
Random variables
- Discrete: Bernoulli, Binomial(n,p) mean np var np(1-p), Geometric (mean 1/p), Poisson(lambda) mean = var = lambda, Hypergeometric, Negative binomial.
- Continuous: Uniform, Exponential(lambda) memoryless mean 1/lambda, Normal N(mu, sigma^2), Gamma, Beta, Chi-square, Student t, Cauchy (no mean), Pareto, Lognormal.
- Expectation is linear (even without independence) E[sum X_i] = sum E[X_i]; Var(X) = E[X^2] - (E X)^2; Var(sum) = sum Var + 2 sum Cov; indicator variable trick.
- Joint, marginal, conditional distributions; covariance and correlation; law of total expectation and variance; transformations (Jacobian).
- Moment generating function M(t) = E e^{tX}; characteristic function; probability generating function.
Limit theorems & inequalities
- Markov P(X >= a) <= E X / a; Chebyshev P(|X - mu| >= k sigma) <= 1/k^2; Chernoff/Hoeffding bounds; Jensen E f(X) >= f(E X) for convex f.
- LLN: sample mean -> mu (weak: in probability; strong: a.s.). CLT: sqrt(n)(mean - mu)/sigma -> N(0,1). Berry-Esseen rate.
- Convergence modes: a.s., in probability, in L^p, in distribution.
- Borel-Cantelli lemmas; Kolmogorov 0-1 law.
Stochastic processes
- Markov chains: transition matrix P, stationary distribution pi P = pi, irreducible/aperiodic -> convergence; absorbing chains; hitting times; gambler's ruin; random walks (recurrent in 1D/2D, transient in 3D, Polya).
- Poisson process (exponential inter-arrivals), birth-death, queues (M/M/1: rho = lambda/mu, L = rho/(1 - rho)), renewal theory.
- Martingales, optional stopping, Doob inequalities; Brownian motion and Ito calculus (Black-Scholes); branching processes.
What ships with it
1 file beside SKILL.md in the same directory: the scripts, references and assets a skill reads on demand. Not counted in the per-session cost; read them before you install if any of them is executable.
What this file has done since we first saw it
Hashed on every crawl. A supply-chain change to an agent config is a question of when, not whether, so the history is kept rather than the latest state alone.
- today Changed · -23 lines · +31 tokens per session 69764f73f75c
- 7d ago First seen · 56 lines · 31 tokens per session scan A 04e327630f08
probability-theory is a skill published in the GitHub repository hamzabellouch/agent-skills (8 stars, last pushed today), licensed MIT. It adds 62 tokens to every session and 728 once invoked, about $0.0002 per session on Opus 5.5. A static security scan graded it A with 0 findings. No closer match exists in the catalogue, so it is treated as the original; first seen 2026-10-10.
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