Solitaire Odds & Statistics
Win rates, probabilities, and the fascinating mathematics that govern every game of Solitaire.
Solitaire is a game where luck and strategy intertwine — and understanding the underlying mathematics can transform how you think about each game. Here are the numbers behind the world’s most popular card game.
Win Rates by Variant
| Variant | Theoretical Win Rate (Optimal Play) | Average Player Win Rate |
|---|---|---|
| Klondike (draw-1) | ~82% | ~15–25% |
| Klondike (draw-3) | ~82%* | ~10–15% |
| FreeCell | ~99.999% | ~75% |
| Spider (1-suit) | ~99% | ~80% |
| Spider (2-suit) | ~50% | ~25% |
| Spider (4-suit) | ~33% | ~10% |
| Pyramid | ~2% | ~1% |
| Tri-Peaks | ~90% | ~80% |
| Canfield | ~35% | ~15% |
| Yukon | ~80% | ~25% |
*Klondike’s theoretical win rate is the same for draw-1 and draw-3 since the cards are the same — only accessibility differs in practice.
The Klondike Numbers in Depth
How 82% Was Determined
Determining Solitaire’s theoretical win rate is an extremely complex computational problem. Unlike FreeCell (where all information is visible), Klondike’s hidden cards make exhaustive analysis difficult.
Key research findings:
- Computer solvers using advanced algorithms have analyzed millions of random Klondike deals.
- The best-known result, from researchers including Yan et al. (2005) and subsequent work, places the theoretical win rate at approximately 79–82% for draw-1 with unlimited passes.
- This means roughly 18 out of every 100 deals are mathematically impossible to win, regardless of how well you play.
Why Average Players Win Far Less
If ~82% of deals are winnable, why do most players only win 10–20%? Several factors:
- Imperfect decisions: With 21 hidden cards, even experienced players make suboptimal moves.
- No take-backs: In physical play (and most digital settings), you commit to decisions without knowing outcomes.
- Limited stock passes: Many rule sets restrict stock recycling, which prevents accessing all cards.
- Difficulty of optimal play: Finding the winning path in a complex deal can require evaluating hundreds of possible move sequences.
The Skill Gap
| Player Level | Approx. Win Rate | Notes |
|---|---|---|
| Random play | 2–5% | Moving cards without any strategy |
| New player | 5–10% | Basic understanding of rules |
| Casual player | 10–15% | Some strategic awareness |
| Good player | 20–30% | Applies strategic principles consistently |
| Expert | 30–40% | Deep understanding, forward planning |
| Computer solver | 70–82% | Exhaustive search algorithms |
The jump from “random play” to “expert” represents a 6–8× improvement — an enormous skill gap for a game many consider purely luck-based.
FreeCell: The Near-Perfect Rate
FreeCell’s ~99.999% winnability is remarkable and has been verified through extensive computational analysis.
The Famous Unsolvable Deals
Among the first 32,000 numbered Microsoft FreeCell deals:
- Deal #11982 — confirmed unsolvable
- Deal #146692 — unsolvable (in extended numbering)
- Deal #186216 — unsolvable (in extended numbering)
Out of 1 million random FreeCell deals tested by various solvers, only ~8–12 were found to be unsolvable. This translates to roughly 0.001% or 1 in 100,000.
Why FreeCell’s Rate Is So High
Three factors combine:
- Complete information: All 52 cards face-up enables perfect planning.
- Flexible storage: 4 free cells + empty columns provide maneuvering room.
- No rank restriction on empty columns: Any card can fill an empty column.
Spider Solitaire Statistics
Spider’s win rate varies dramatically by difficulty mode because the number of suits directly determines how hard it is to complete same-suit sequences.
| Mode | Suits | Approximate Win Rate |
|---|---|---|
| 1-suit | 1 (Spades) | ~99% |
| 2-suit | 2 (Spades, Hearts) | ~50% |
| 4-suit | 4 (all) | ~33% |
Why 1-Suit Is So Easy
When only one suit exists, every descending sequence is automatically a same-suit sequence. You never waste moves building mixed-suit stacks. The game reduces to a simple ordering puzzle.
Why 4-Suit Is So Hard
With all four suits, building a complete same-suit K→A run requires finding 13 specific cards of one suit among 104 total cards across 10 columns. Mixed-suit stacks form frequently and must be carefully dismantled.
The Mathematics of Shuffling
How Many Possible Deals Exist?
For a standard 52-card deck:
$$52! = 80{,}658{,}175{,}170{,}943{,}878{,}571{,}660{,}636{,}856{,}403{,}766{,}975{,}289{,}505{,}440{,}883{,}277{,}824{,}000{,}000{,}000{,}000$$
That’s approximately $8.07 \times 10^{67}$ — a number so large it defies intuition.
Putting It in Perspective
- There are approximately $10^{80}$ atoms in the observable universe.
- There are approximately $10^{67}$ possible Solitaire shuffles.
- Every game you play is almost certainly unique — the probability of randomly dealing the same arrangement twice in a human lifetime is effectively zero.
For Spider (2 Decks)
Spider uses 104 cards (two combined 52-card decks). The number of distinct arrangements is:
$$\frac{104!}{(2!)^{52}}$$
This accounts for the fact that pairs of identical cards are interchangeable. The result is still an astronomically large number.
Probability of Specific Card Positions
Ace on Top of First Column (Klondike)
The probability of the first column’s face-up card being an Ace:
$$P = \frac{4}{52} = \frac{1}{13} \approx 7.7%$$
At Least One Ace in Starting Face-Up Cards
With 7 face-up cards starting, the probability of at least one being an Ace:
$$P = 1 - \frac{\binom{48}{7}}{\binom{52}{7}} \approx 43%$$
So you’ll start with at least one visible Ace in roughly 4 out of 10 games.
All Four Aces in Starting Face-Up Cards
$$P = \frac{\binom{4}{4} \times \binom{48}{3}}{\binom{52}{7}} \approx 0.13%$$
An extremely rare starting position — about 1 in 750 games.
Draw-1 vs. Draw-3 Probability Analysis
Card Accessibility
- Draw-1: Every card in the 24-card stock is accessible one at a time. With unlimited passes, you can reach any card.
- Draw-3: On each pass, you can access cards at positions 3, 6, 9, 12, 15, 18, 21, 24 — only every third card.
This means draw-3 makes approximately two-thirds of the stock inaccessible per pass, drastically reducing your options compared to draw-1.
Impact on Win Rate
While the theoretical maximum win rate is similar (the same deals are winnable), practical win rates differ significantly:
| Mode | Average Player Win Rate |
|---|---|
| Draw-1, unlimited passes | ~20–25% |
| Draw-1, 1 pass | ~15% |
| Draw-3, unlimited passes | ~12–15% |
| Draw-3, 3 passes | ~8–10% |
Interesting Statistical Facts
- Microsoft reports that the average Solitaire Collection user plays about 100 games per month.
- Completion rates on Microsoft Solitaire Collection hover around 33% for Klondike (assisted by hints and unlimited undo).
- The longest theoretical Solitaire game (maximum moves before winning) involves approximately 200+ individual card moves.
- The shortest possible win requires about 52 moves in an ideal deal (moving each card exactly once to its foundation).
- Clock Solitaire has a precise win rate of $\frac{1}{13} \approx 7.7%$ — the probability that all four Kings aren’t exposed before the rest of the clock is complete.
What Statistics Tell Us About Strategy
Understanding the numbers reveals strategic truths:
- Don’t blame luck entirely — since ~82% of deals are winnable, most losses involve some suboptimal play.
- Improvement is real and measurable — moving from 10% to 30% win rate is achievable through better strategy.
- Some losses are inevitable — even perfect play can’t beat ~18% of deals. Don’t let impossible games discourage you.
- Choose your difficulty — if winning more often matters to you, play draw-1 or try 1-suit Spider.
For strategies to improve your numbers, see Solitaire Strategy and Tips for Winning.
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