Separate a probability from a proof
A proved safe square has zero mine risk under the established rules and constraints. A square with a low estimated risk remains a guess. Keeping these categories separate helps you judge the quality of a move after the outcome: a careful 10% risk can hit a mine, and an unsupported choice can happen to survive.
Start with deductions. Scan other frontiers, compare overlapping clues, and use the remaining mine count before estimating odds. If a safe action can reveal more information, the uncertain position may become solvable without any guess.
Probability is appropriate when the information leaves several plausible arrangements. It is not a shortcut for skipping neighbor counts, and it cannot repair an inconsistent collection of speculative flags. Before trusting a probability, state what assumptions and evidence went into it.
When mines divided by squares is useful
For an interchangeable set of unknown squares with exactly M mines and N squares, each square has mine probability M/N. One mine uniformly placed among five unconstrained squares gives each a 20% risk; four among twenty also gives 20%.
The interchangeability assumption is doing the work. A frontier clue may force one covered square to be a mine and another to be safe, even though the board-wide density is 20%. Applying the overall percentage to both ignores the evidence.
Use the ratio as a baseline for an outside area only when its mine count is known and its squares are not distinguished by additional information. If the mine count outside varies between frontier arrangements, calculate a weighted average over those arrangements instead.
Starting density is not current risk. Expert begins with 99 mines among 480 squares, but after you reveal safe cells and identify mines, the unresolved board has a different distribution. A raw starting percentage cannot evaluate a specific late-game move.
List the valid arrangements before assigning odds
Suppose the visible constraints are A + B = 1 and B + C = 1. There are two frontier arrangements: B is a mine and A and C are safe, or A and C are mines and B is safe. Each satisfies both local clues.
It is tempting to assign each branch 50% because there are two. That is only justified if the branches have equal total probability weight. They use different numbers of mines, so the global mine count and outside area may make them unequal or eliminate one entirely.
Under a model that gives equal weight to every complete remaining mine placement, count how many complete boards each branch represents. If there are three outside squares and two remaining mines, the B-only branch needs one outside mine. That mine can occupy any of the three squares, giving three completions. The A-and-C branch needs zero outside mines, giving one completion.
There are four equally weighted complete arrangements in total. B is mined in three, so its risk is 75%. A and C are each mined in one, so each has a 25% risk. The two local branches were not equally likely.
Count arrangements, not just branches
This is a constraint diagram, not a spatial board. One clue says A + B = 1; another says B + C = 1. The symbols name covered squares. No other clues restrict them.
| Frontier mines | Outside mines needed | Completions | Weight |
|---|---|---|---|
| B | 1 | 3 | 75.0% |
| A + C | 0 | 1 | 25.0% |
Choose A, B, or C to inspect it.
This model gives equal weight to complete placements of the remaining mines. Outside squares have no additional clues. It illustrates conditional probability; it does not solve your current game.
Why outside completions change the weights
The number of ways to put k mines in n otherwise unrestricted squares is the binomial coefficient “n choose k.” For three squares, there is one way to place zero mines, three ways to place one, three ways to place two, and one way to place all three.
In the explorer, each branch leaves a different k for the outside area. A negative k is impossible. A k larger than the number of outside squares is also impossible. Those branches receive zero weight rather than a small probability.
With three outside squares and three mines remaining, the B-only branch leaves two outside mines and has three completions. The A-and-C branch leaves one and also has three. In that model the branches are equally weighted, and A, B, and C each have 50% mine risk—even though their mine occurrences are linked in different ways.
These calculations assume no additional outside clues and equal weighting of complete placements. A particular game’s generation procedure can affect the appropriate probability model. The explorer illustrates the calculation under its stated model; it does not claim to be a live solver for Minesweeper Now boards.
Check your understanding
One frontier branch has three equally likely complete-board completions; another has one. What is the first branch’s probability?
Choose an answer.
What makes a real 50/50?
A genuine 50/50 has two alternatives with equal probability weight after all available information is included. A familiar example is an isolated pair containing exactly one mine, with no clue or global constraint distinguishing the two placements under a symmetric model.
Two covered squares are not automatically a 50/50. They might both be safe, both be mines, or have unequal odds because other clues distinguish them. A visually symmetric local pair can also become asymmetric when the rest of the board constrains how many mines are available.
Distinguish a currently ambiguous pair from an unavoidable guess. If there is a safe reveal elsewhere that could produce a distinguishing clue, solve that region first. You have not exhausted the available information while a useful guaranteed-safe action remains.
When the ambiguity truly cannot be resolved, do not blame yourself for a mine hit merely because the other square would have survived. Evaluate whether the alternatives were correctly identified and whether a safer move was overlooked elsewhere.
Immediate safety and information are different goals
A move with the highest immediate survival probability is not necessarily the move with the highest chance of eventually winning. Surviving one reveal may leave another difficult guess; a slightly riskier reveal may expose information that resolves several areas.
For a simple illustration, a route requiring two independent 80%-safe guesses has a 64% chance of surviving both: 0.8 × 0.8. A different route that has a 70%-safe opening and then guarantees the rest would have a 70% completion chance. The higher-risk first move would be preferable in that hypothetical comparison.
Real positions are more complicated. Successive guesses are often dependent, the information revealed varies, and the later route may not be known. Do not multiply percentages as though moves were independent unless the model supports that assumption.
A practical approach is to prioritize a clearly safer move, then consider information when the risks are similar. Favoring a move because it “might open a lot” is not proof that it improves completion odds. State the tradeoff honestly rather than assigning an invented exact percentage.
Review uncertain moves without outcome bias
- Record whether a move was forced or uncertain before judging the result.
- Identify the clues and mine-count constraints you actually used.
- Check for safe moves elsewhere that you missed.
- If you estimated odds, verify whether branches were weighted correctly.
- Consider what useful information a surviving reveal could produce.
- Judge your decision from the information available at the time, not from the hidden answer revealed afterward.
Can a solver always tell me the exact odds?
A solver can compute probabilities under a chosen board model if it accounts for the relevant constraints and mine budget. Its output is only as appropriate as that model and the information supplied. A misplaced assumed flag can invalidate the calculation.
Are corners always the safest guesses?
No. A corner has fewer neighbors, but its risk still depends on the clues and valid mine arrangements. The shape alone does not establish a universal probability advantage.
Should I flag a square that is probably a mine?
Keep uncertain annotations separate from proved flags. If you later use an uncertain flag to satisfy a clue or subtract from the mine total, a probability can silently become a false premise.