“Random” describes how a deal is selected. “Winnable” says that a legal winning sequence exists. They are different properties: a game can choose randomly from a pool containing only deals already proved winnable.
Method: a mathematical explainer using a made-up collection of ten deals. No player records were collected and no new solver experiment was run. The purpose is to show why completion rates from different deal-selection systems cannot be compared as if everything else were equal.
A ten-deal example you can check by hand
Suppose you have ten deals, each with the same chance of being chosen. Eight have a solution and two do not. A particular playing method completes four of the eight solvable deals and none of the two impossible ones.
| Selection rule | Possible deals | Deals this method completes | Completion probability |
|---|---|---|---|
| Uniform choice from all ten | 10 | 4 | 4/10 = 40% |
| Uniform choice from the eight solvable deals | 8 | 4 | 4/8 = 50% |
The playing method did not improve. Its winning percentage rose because the two impossible deals were removed from the count. Even in the second row, half the attempts fail: guaranteeing that a solution exists does not guarantee that the player finds it.
In this simplified model, write S for solvability and W for completion. Because a genuine win requires a solvable deal, P(W) = P(S) × P(W | S). Here, 0.8 × 0.5 = 0.4. Choosing only solvable deals gives the second row. In plain numbers: eight of ten deals are solvable, and the method wins half of those eight, so it wins four of ten overall.
A computer-checked collection may leave out other winnable deals
A solver is a program that searches for a winning sequence of moves. It may find some solutions and run out of time on other deals. A collection containing only its confirmed wins can therefore leave out deals that are also winnable. It may include some kinds of deal more often than others.
Suppose the solver confirms four of the eight solvable deals. If our playing method succeeds on three of those four, the selected-pool completion rate is 75%. If it succeeds on only one, the rate is 25%. Both are compatible with the same 40% rate across the full ten-deal population. Changing the collection can make the same playing method look better or worse.
The 2024 version of The Winnability of Solitaire and Patience Games studies winning-sequence existence under specified game rules. Its subject is distinct from recording whether ordinary players finish. Our methodology’s evidence labels likewise separate proved results, lower bounds and unresolved searches.
Check what “random” means in that game
A uniform mathematical shuffle assigns equal probability to every ordering of a deck. Software may instead use a seeded generator or select a deal from a finite catalogue. The word “random” alone does not tell you which deals are included, whether each has an equal chance, or how well the selection works.
This article does not test the site’s generator for uniformity. The opening-Ace calculation explicitly assumes the ideal uniform model. Its exact result should not be mislabeled as a measured frequency from a particular app.
Compare results from the same kind of deal
Before comparing two percentages, record the draw mode, permitted stock passes, deal pool, solver rules and whether undo or replay is included. A certification applies to a particular ruleset; changing that ruleset can change whether the recorded solution remains legal.
When you play Solitaire and compare your results, keep the selected mode with your notes. Use the published report for its defined groups of recorded games, and the calculation script to reproduce the fractions in this article’s made-up example. A high completion rate is useful information only when you know what was being selected and counted.