Fair multipliers vs house-edge multipliers in mines

In mines, the fair multiplier for any number of picks is 1 ÷ P, where P is the chance of surviving those picks; a game with a house edge e pays (1 − e) ÷ P instead. With 3 mines and 5 picks, P = 49.57%, so the fair multiplier is ×2.0175 and a 1%-edge game pays about ×1.9974.

That small gap is how the game makes money, and it's present at every pick count and mine count. This guide shows how to compute both prices and how to read a game's edge straight off its multiplier table.

Why 1 ÷ P is "fair"

A bet is fair when the average payout equals the stake. If you win with probability P and receive multiplier M when you do, the average return per unit staked is P × M. Setting P × M = 1 gives M = 1 ÷ P.

Example: three picks with 3 mines survive 66.96% of the time. 1 ÷ 0.6696 = ×1.4935. Over a very long run of 1-unit rounds, you'd get back 1 unit on average for every unit staked — no gain, no loss.

How the edge is applied

Real games don't pay the fair price. They multiply it by (1 − e). The average return becomes P × (1 − e) ÷ P = 1 − e, whatever P is. That's the key point: the edge is the same for every choice on the board.

fair multiplier      M_fair = 1 / P
edge multiplier      M_edge = (1 - e) / P
expected return      P × M_edge = 1 - e      (for every m and k)

Table: 3 mines

Survival probability from the probability tables, then three prices.

Table: 5 mines

The gap between fair and edge prices grows in absolute terms as multipliers get bigger (0.0114 at one pick with 3 mines, 0.0859 at eight picks with 5 mines), but as a percentage it's always the same 1%.

Worked example: reading a game's edge (fictional)

Fictional figures for illustration — not taken from any real game. Suppose a mines game shows these multipliers with 3 mines:

Each P × multiplier is the expected return per unit staked. Here it's around 0.98, so the edge is roughly 1.4–1.9%, with the spread caused by rounding to two decimals. A game that rounds multipliers down to two decimal places quietly adds a little to the edge on top of its headline figure.

What the edge costs over many rounds

Because the expected return is 1 − e on every round, the expected cost of playing is easy to estimate: rounds × stake × edge.

That's an average. Individual sessions swing well above and below it, and the swings get wider as multipliers get bigger. But no setting moves the average in your favour.

Rounding and minimum steps

Multipliers on screen are usually rounded, often to two decimal places. If a game rounds down, the real edge is a little higher than the headline figure, and the difference is largest on small multipliers (where 0.01 is a bigger share of the payout). Some games also apply a minimum multiplier for the first pick. Both effects show up if you work out P × multiplier for several pick counts, as in the example above.

Why bigger multipliers aren't better value

It's tempting to think a ×17.5 payout must be a better deal than ×1.24. It isn't. The ×17.5 comes with a 5.65% chance; the ×1.24 comes with 80%. Multiply each pair and you get 0.99 both times. The only thing that changes is how bumpy the ride is: high multipliers mean long runs of losses broken by occasional big wins, low multipliers mean frequent small wins broken by a loss. Over time, both drain at the same 1% rate. We test this with simulations in expected value: why no pick count beats the edge.

Check it yourself

from math import comb
def p_survive(m, k): return comb(25 - m, k) / comb(25, k)

m, edge = 3, 0.01
for k in (1, 3, 5, 10):
    p = p_survive(m, k)
    fair, paid = 1 / p, (1 - edge) / p
    print(k, round(p, 4), round(fair, 4), round(paid, 4), round(p * paid, 4))
# 1 0.88 1.1364 1.125 0.99
# 3 0.6696 1.4935 1.4786 0.99
# 5 0.4957 2.0175 1.9974 0.99
# 10 0.1978 5.0549 5.0044 0.99

The browser mines calculator uses the same formula and lets you set any edge.

Multiplier checklist

• Note the mine count and pick count for the multiplier you're looking at.
• Look up or compute P = C(25 − m, k) ÷ C(25, k).
• Multiply P by the shown multiplier. That's the expected return per unit.
• 1 minus that number is the effective edge, including rounding.
• Check a few pick counts; they should give roughly the same edge.
• Remember: any result below 1 means you should expect to lose over time.

Summary

Fair multipliers are just probabilities turned upside down. The house edge shaves every one by the same percentage, so the expected return is identical across the whole board. Knowing this won't help you win; it helps you see exactly what each round costs. For the game rules, see how the mines game works.