Mines odds explained: the maths behind the 5×5 grid

The setup

Twenty-five tiles. Some number of mines, hidden uniformly at random. In LarpCasino the default is 3 mines and the range is 1 to 24. You reveal tiles one at a time, and each safe reveal raises the multiplier you would be paid if you cashed out now.

There is no skill in tile selection. Every unrevealed tile has the same probability of hiding a mine, so corners, edges, patterns and hunches are all identical bets. What you actually control is one thing: when to stop.

Survival probability, step by step

With m mines on 25 tiles, the first pick is safe with probability (25 − m) / 25. If it was safe, 24 tiles remain and m of them are mines, so the second pick is safe with probability (24 − m) / 24. And so on.

Surviving k picks means every one of those independent-looking conditional steps went your way, so you multiply them together. With 3 mines:

Two things are visible in that table. Survival collapses far faster than intuition suggests — on the default three mines, you fail to bank a fifth gem slightly more often than you manage it. And the fair multiplier is exactly the reciprocal of the survival chance, which is the definition of a fair price.

Where the house edge enters

A fair game pays 1 / P(survive). A commercial one pays slightly less. In LarpCasino the relationship is stated outright: the paid multiplier is (1 − edge) × fair.

Multiply that through and the consequence is exact: survival chance × paid multiplier = 1 − edge, at every rung of the ladder. Cashing out after one gem and cashing out after twenty have identical expected value.

What the mine count actually changes

Raising the mine count makes each pick more dangerous and each gem worth more, and those two effects cancel in the same way. With 24 mines on the grid there is a single safe tile, so one correct pick pays 25× before the edge, and there is no second pick to make.

So the mine slider is a volatility control with a very wide range. It shapes the distribution of outcomes dramatically and moves the long-run return by exactly nothing.

Common misreadings

1. 01"Corners are safer." The mines are placed uniformly at random. Position carries no information, and neither does the order in which you click.
2. 02"The grid is due." Each round is a fresh placement. A run of early busts says nothing about the next round.
3. 03"Cash out at three every time and you grind up." You grind up most rounds and give it back on the busts. Expected value is 1 − edge, the same as any other stopping rule.
4. 04"More mines means better value." More mines means bigger multipliers priced at exactly the risk you took on. Value is constant.

Checking it instead of trusting it

Mines is the game where the identity survival × multiplier = 1 − edge is easiest to verify, because the rungs are discrete and the game shows you the multiplier before you commit to each one.

In LarpCasino — a simulated casino for iPhone, with invented credits and no real money at any point — you can do that directly. Set the edge to 0% and check that every rung's multiplier is the reciprocal of its survival chance. Set it to 10% and watch the entire ladder shift down by the same factor. Then run the RTP simulator over 100,000 or 1,000,000 rounds and confirm the empirical return lands on 1 − edge regardless of where you were stopping.

If the arithmetic above is new, house edge explained covers the general case, and how crash games work shows the same cancellation in a continuous game rather than a discrete one.

The short version

• All unrevealed tiles are identical. Tile choice is not a decision; stopping is.
• Survival after k picks is a product of shrinking conditional probabilities, and it falls quickly.
• Fair multiplier is 1 / P(survive); the paid multiplier is that scaled by 1 − edge.
• Every cash-out rung has the same expected return. Depth sets volatility only.
• The mine count is a volatility dial with a very wide range and no effect on long-run return.