Reconstruct the House Edge From a Plinko or Mines Payout Table
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Plinko and Mines publish everything you need to work out exactly what they cost you, and almost nobody does the arithmetic. Both games show a payout table — the multipliers down the sides of the Plinko pyramid, the rising payout per safe tile in Mines — and both have a structure simple enough that the house edge follows from the table by hand.
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This is unusual. A slot's edge is buried in a provider's maths document; here it is on screen, if you are willing to multiply.
On this page4
The principle in one line
For any single bet, the expected return is the sum of each outcome's payout multiplied by its probability. Divide that by the stake and you get the return to player; subtract from one and you have the house edge. Everything below is that one calculation applied twice.
Plinko: a binomial in disguise
A ball dropped through a pyramid of pegs takes a left or right deflection at each row. With n rows, the number of distinct landing slots is n+1, and the probability of landing in a given slot is the binomial distribution with p = 0.5. For a 16-row board, the probability of landing in slot k (counting from 0) is C(16,k) divided by 216, which is 65 536.
The binomial coefficients for 16 rows are the 17 numbers 1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1. They sum to 65 536. The outermost slot — the one carrying the big multiplier the game is marketed on — is reached once in 65 536 drops. The centre slot, carrying a multiplier below 1, is reached 12 870 times in 65 536, which is a little under one drop in five.
Doing it with your own board
Write the game's 17 multipliers in order. Multiply each by its coefficient from the list above. Add the products. Divide by 65 536. That is your return to player, as a multiple of stake, for that risk setting on that board.
Two things fall out immediately. First, the headline multiplier at the edge contributes almost nothing: even a 1000× payout reached once in 65 536 drops adds about 0.015 to the return. Second, almost the whole return comes from the five or seven central slots, where the multipliers are below 1. The game's economics live in the boring middle, which is the opposite of where the interface points your attention.
Run the same calculation on the low-risk and high-risk tables of the same board and compare. In most implementations the returns come out close to each other — the risk setting is a variance control, not a value choice — but not always identical, and where they differ it is worth knowing which setting is the cheaper one. An 8-row board uses the coefficients 1, 8, 28, 56, 70, 56, 28, 8, 1 over 256; a 12-row board uses 1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1 over 4096.
Mines: a chain of conditional probabilities
Mines is a 5×5 grid of 25 tiles with m mines placed at random. You open tiles one at a time; each safe tile raises the multiplier, and the first mine ends the round. The probability of surviving k picks with m mines on the board is a product of fractions:
(25−m)/25 × (24−m)/24 × (23−m)/23 × … for k terms.
With 3 mines, surviving one pick is 22/25 = 0.88. Surviving two is 22/25 × 21/24 = 0.77. Surviving three is 0.77 × 20/23 = 0.669. Surviving five is about 0.496 — slightly worse than a coin flip, which is a useful calibration against how safe five picks feels.
Reading the edge off the cashout button
The game shows you the multiplier for cashing out after each successful pick. Multiply the multiplier offered after k picks by the survival probability for k picks. The result is your expected return for that plan. Do it for several values of k and you will find the same number coming back each time, within rounding — that constant is 1 minus the house edge, and its constancy is the signature of a correctly built game.
A fair-with-no-edge game would offer, after three picks with three mines, a multiplier of 1 ÷ 0.669 = 1.495×. If the button says 1.47×, the edge is 1 − (1.47 × 0.669) = about 1.7 %. That is the whole calculation, and it takes less time than a round does.
Two refinements worth making. Compare across mine counts: a board set to 1 mine and a board set to 10 should return the same constant, and if one is noticeably worse, that configuration is the expensive one. And check the extremes — the multiplier for opening every remaining tile sometimes carries a worse rate than the mid-range, because that is the number used in the marketing.
What the exercise is good for
Not for finding a winning setting. Both games have a positive house edge at every setting that matters, and the point of the arithmetic is to find out how large it is at the specific site you are using, which varies more than you would expect for games that look identical. A game advertising 1 % and delivering 1 % is doing what it says. A game advertising 1 % whose payout table works out to 4 % is telling you something about the operator rather than about the game.
It is also the cleanest demonstration available that provably fair and fair-odds are different claims. A provably fair Plinko proves the ball's path was not altered after your bet. It says nothing about whether the multiplier table on the side of the pyramid is generous or brutal — that is a separate question, answerable only by multiplying out the table yourself.
Gambling is for adults and the arithmetic above is the reason losing is the expected outcome rather than the unlucky one. If play has stopped being a choice, the services listed on our responsible gambling page are free and independent of any operator.