# Pocket aces waiting-time distribution lab

Generated and independently checked on 2026-09-11 for the GTO Gecko article “How Long Until Pocket Aces?”. The bundle uses no private hand histories, product data, user data, or random simulation.

## Reproduce

```text
node generate.mjs
python verify.py
python -O verify.py
```

`generate.mjs` derives each event from exact card combinations, evaluates geometric first-hit probabilities and negative-binomial repeat waits, and writes both localized SVGs. `verify.py` is an independent Python implementation using `fractions.Fraction`, `math.comb`, direct binomial sums, full-row checks for all three CSVs, and a manifest check.

## Model

- Standard 52-card Texas Hold’em deck, two private cards, one fixed seat.
- Every hand is freshly and independently shuffled.
- A wait includes the successful hand. The model counts hands, not minutes or sessions.
- “Pocket pair, then at least one matching rank in a hypothetical flop runout” deals three community cards every hand, whether or not live action would reveal a flop. It includes a set, a full house on a paired flop, or quads; it is not an observed-flop rate or set-mining profitability test.
- Point frequency and waiting time are not strategy EV, proof of shuffle fairness, or a promise about the next hand.

For an event with per-hand probability `p`, the probability of seeing it at least once within `n` hands is `1 - (1 - p)^n`. The mean first-hit wait is `1 / p`. Integer quantiles are the first hand count whose cumulative probability reaches the requested threshold. This release includes the successful hand; references that define a geometric variate as failures before success are therefore one count lower. For the `r`th occurrence, `P(T_r <= n) = P(Binomial(n, p) >= r)`.

## Files

- `experiment.json`: assumptions, exact counts, equations, event metrics, drought checkpoints, and repeat waits.
- `event-waits.csv`: first-hit summary for six precisely scoped events.
- `aa-cdf.csv`: complete 0–1,000-hand pocket-aces cumulative distribution.
- `aa-repeat-waits.csv`: mean, median, 90%, and 95% waits for 1, 5, 10, and 20 AA occurrences.
- `waiting-curve.svg` / `waiting-curve-ja.svg`: localized charts generated from the same AA distribution.
- `generate.mjs` / `verify.py`: generator and independent verifier.
- `MANIFEST.sha256`: hashes for the public release files.

## Source route

Poker rules and probability definitions were checked on 2026-09-11 against PokerStars’ current Texas Hold’em rules, Stanford CS109’s geometric/negative-binomial teaching material, Penn State STAT 414 Lesson 11, and NIST’s geometric-distribution documentation. The calculations in this bundle are independently derived; those sources define the game and mathematical model rather than supplying the reported poker results.