# Prospect 51: exact order-run odds before and after a hold

**Site research addendum · 30 September 2026.** This note analyzes the declared v3.5 Prospect 51 rule. It does not change the game, the frozen Project 51 paper, or a DOI deposit. Its claims are exact for the model below; they are not measured win rates from visitors.

## Model and opening probability

The deck has 51 distinct identities: the usual 52 cards without 2♣. Five ordered rotors deal without replacement. At modulus \(m\), the rotor takes an independent uniform digit \(d\), applies \(e=(ad+b)\bmod m\) with \(\gcd(a,m)=1\), and removes position \(e\). Thus every ordered opening hand has probability \(1/(51\cdot50\cdot49\cdot48\cdot47)=1/281{,}887{,}200\). The result assumes independent uniform input digits; the choice of rotor wiring and knob does not alter it.

An order run uses five consecutive ranks in strict left-to-right ascending or descending order. Ace may be low in A–2–3–4–5 or high in 10–J–Q–K–A. There are ten ascending rank patterns and their ten reverses. Two patterns contain a two and have \(3\cdot4^4=768\) identity assignments each, because 2♣ is missing. Each of the other eight has \(4^5=1{,}024\). Hence the number of qualifying ordered opening hands is

\[
2\bigl(2(3\cdot4^4)+8(4^5)\bigr)=19{,}456,
\qquad
\Pr(\text{opening order run})=\frac{19{,}456}{281{,}887{,}200}
=\frac{608}{8{,}808{,}975}\approx0.00690205\%.
\]

That is about one in 14,488 ordered openings. It is **not** the rate of order runs after a player chooses holds.

## A hold changes the sample space

Let \(I=(I_1,\ldots,I_5)\) be the initial ordered hand and \(H\) the set of held positions. The \(r=5-|H|\) replacements come from the 46 identities outside *all* five initial cards; discarded initial cards do not return. Let \(c_q(I)\) be the number of those 46 identities of natural rank \(q\), and let \(\mathcal L\) be the 20 allowed ordered rank patterns (Ace has natural rank 1 in both Ace-low and Ace-high patterns). The exact conditional probability is

\[
\Pr(\text{run}\mid I,H)=
\frac{1}{(46)_r}
\sum_{\substack{\ell\in\mathcal L\\
                 \operatorname{rank}(I_i)=\ell_i\ \forall i\in H}}
       \prod_{i\notin H}c_{\ell_i}(I),
\qquad (46)_r=\frac{46!}{(46-r)!}.
\]

For each compatible pattern, its five ranks are distinct, so the product counts its ordered identity completions without overlap. For example, from 5♥–6♠–7♦–8♣–9♥, holding positions 1, 2, 3 and 5 while replacing the 8 gives exactly three successful cards among 46: a run probability of \(3/46\). Holding all five makes the run certain.

The game's existing hold-odds panel counts \(\binom{46}{r}\) unordered completions for the **base poker award**, which ignores order. Order-run odds instead use \((46)_r\) ordered completions. The archive-lock jackpot is fixed by the original order and does not change with the hold.

## Expected bonus, with scope

The triggered bonus is one free five-card spin from a fresh 51-card inventory. It pays 5 jbits plus that spin's ordinary poker gross award, with no entry, hold, or chained bonus. An independent full \(\binom{51}{5}=2{,}349{,}060\)-hand census using the v3.5 paytable totals 802,746 gross jbits. Under independent uniform digits, the fresh spin's expected poker gross is

\[
G=\frac{802{,}746}{2{,}349{,}060}
=\frac{19{,}113}{55{,}930}\approx0.341730735\text{ jbits}.
\]

Consequently, the expected **gross bonus per triggered run** is \(5+G=298{,}763/55{,}930\approx5.341730735\) jbits. For a specified initial hand and hold, its expected bonus contribution is \(\Pr(\text{run}\mid I,H)(5+G)\), provided the fresh spin is independent of the triggering hand. Add this to the base-award expectation when comparing holds. The already-paid 1-jbit entry and opening-order archive prize are constant across those choices.

This does not establish an optimal hold policy or a profitable game. As a simple check, a paying opening hand J♠–J♥–3♦–7♣–9♠ gives 1 jbit if kept, while holding the two jacks and replacing three has exact base gross expectation \(23{,}440/15{,}180=1.544137\ldots\) jbits. The “auto hold” setting is a convenience, not a strategy recommendation.

## Replay

Run `python research/project-51/prospect-51/verify_order_run_2026_09_30.py` from the site root. The self-contained verifier independently scores all 2,349,060 five-card subsets, derives the opening count, and checks the conditional formula against direct ordered redraw enumeration on seven cases, including Ace-high. It compares its output byte for byte with [the receipt](order_run_receipt_2026_09_30.json). Its source is [here](verify_order_run_2026_09_30.py).

The rule audited is [Prospect 51 v3.5](../../../area-51/prospect-51/) with its [frozen engine](../../../area-51/prospect-51/engine-v2-7.js). The verifier imports neither engine code nor its evaluator.