Expected value in mines: why no pick count beats the house edge

Expected value (EV) is the average result per round over a very long run, and in mines it's the same for every choice: with a house edge e, each 1-unit round is worth −e on average, whether you pick 1 tile with 1 mine or 8 tiles with 5 mines. No pick count, mine count or cash-out point beats the edge, because every multiplier is priced from the same formula.

Below is the short algebra, then a simulation of 20,000 sessions for five different settings to show what does and doesn't change.

The algebra in three lines

Let P be the chance of surviving your planned picks and M the multiplier you'd cash out at.

EV per 1-unit round = P × M − 1
game pays           M = (1 − e) / P
so                  EV = P × (1 − e) / P − 1 = −e

P cancels. That's the whole story: the mine count and pick count only change P, and P drops out. With a 1% edge, EV is −0.01 units per round for every option. The multiplier guide shows the full pricing tables.

Worked EV examples

Over 200 rounds of 1 unit, the expected result is −2.00 units for every row.

The simulation

We simulated 20,000 sessions of 200 rounds each, staking 1 unit per round, for each setting above, using a 1% edge. Each round either survives (with the exact probability) and pays the multiplier, or loses the stake. Code and seed are at the end.

How to read it

• The mean column sits close to −2.00 every time. The small differences are sampling noise; the high-risk settings are noisier, so their means wander more from run to run.
• The "ahead" column rises with risk. That isn't better value. Risky settings stretch results out in both directions, so more sessions finish a little ahead and the losing sessions lose far more.
• The worst-5% column shows the cost of that spread: a bad session at 5 mines / 8 picks loses 64 units or more out of 200 staked, versus about 7 at 1 mine / 1 pick.

In other words, your setting chooses the shape of your results, not their average. The average is fixed by the edge.

A check with real tile placement

To be sure the shortcut "win with probability P" matches an actual board, we also placed mines at random on 25 tiles and picked random tiles, 300,000 rounds per setting:

Both land within sampling error of the exact values and of the expected 0.99 return. Tile choice didn't matter, as the probability guide predicts.

Why "systems" don't work

Fictional examples of common ideas, and what the maths says:

• "Cash out after two picks every time." That's one row in the table above: EV −0.01 per round.
• "Double the stake after a mine." Each round still has EV −0.01 × stake. Bigger stakes make the expected loss bigger, and a losing run can exhaust a budget fast.
• "Switch to 1 mine after a big win." Changing settings changes the variance, not the edge.
• "Avoid the corners." Every hidden tile has the same chance of being a mine. See mines myths.

Variance: the number that does change

If EV is fixed, what are you actually choosing when you pick a setting? Variance — how far a single round's result tends to land from the average. One way to measure it is the standard deviation per 1-unit round (computed exactly, 1% edge):

Higher variance means results take longer to settle near the average, which is why high-risk settings can feel like they're "working" for a while. Run enough rounds and the −1% pull wins out.

What EV is useful for

EV turns the edge into a cost you can plan around. If you play 300 rounds of 1 unit at a 1% edge, expect to lose about 3 units on average, and be ready for a session that goes much worse. A useful habit is to set the session budget first, then treat the expected loss as the price of the entertainment — and stop when the budget is gone.

Session-planning checklist

• Rounds planned × stake × edge = expected cost. Is that a cost I'm happy to pay?
• Have I looked at the worst-5% figure for my setting, not just the average?
• Am I using a fixed stake, not raising it after losses?
• Have I set a time or round limit?
• Do I know where to get help if it stops being fun? (Responsible gambling)

Reproduce the simulation

from math import comb
import random

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

random.seed(2026)
SESSIONS, ROUNDS = 20000, 200
for m, k in ((1, 1), (3, 2), (3, 5), (5, 8), (10, 3)):
    p = p_survive(m, k); mult = 0.99 / p; results = []
    for _ in range(SESSIONS):
        wins = sum(1 for _ in range(ROUNDS) if random.random() < p)
        results.append(wins * mult - ROUNDS)
    results.sort()
    print(m, k, round(sum(results) / SESSIONS, 2),
          round(sum(r > 0 for r in results) / SESSIONS * 100, 1),
          round(results[int(SESSIONS * .05)], 2), round(results[int(SESSIONS * .95)], 2))

Your exact figures will vary slightly with a different seed; the means will stay close to −2.00.