Dice, Crash and Mines Odds

The common published rule takes the random number, multiplies it by 10,001, discards the fraction and divides by 100, so every roll from 0.00 to 100.00 is equally likely. That is 10,001 possible rolls, one more than a hundred-point scale with two decimals would suggest. The player picks a target, wins when the roll lands under it, and is paid 99 divided by the target, as if the target were the exact percentage chance of winning.

That price is slightly off. The winning rolls number the target times 100, but they sit among 10,001 outcomes rather than a round ten thousand, so every bet wins a fraction less often than the screen implies. Multiply the payout by the real chance and the house keeps 1.0099% of each stake, not the 1% the game advertises.

The key number: Dice described as a 1% edge game carries 1.0099% in its common published form, because the payout is priced on a hundred-point scale while the roll has 10,001 outcomes.

The real chance counts the winning rolls out of all 10,001. Payouts are rounded down for display, as most dice games show them, which is why a target of 50.5 carries a slightly larger edge than the targets around it.

Limbo draws a multiplier and pays the bet if the multiplier reaches a target chosen beforehand. In the common published form the multiplier is 0.99 over the random number, so the chance of reaching any target is 0.99 divided by that target. Crash shows a multiplier climbing until it busts and pays whoever cashed out before that moment; its hash-chain formula produces the very same curve.

A crash player who always cashes out at 2x is therefore making the same bet as a limbo player who always targets 2x, only spread over a longer round. Cashing out by hand adds one difference: the delay between the click and the server, which can land a cash-out below the intended point or after the bust.

The bottom of the curve is where crash is most often misread. A round that busts at 1.00x, the lowest value the formula allows, pays nobody, and that happens in 1.98% of rounds, about one in 50. It is not a charge on top of the edge but the same curve read at its lowest step, and limbo produces 1.00x results at the same rate.

The figures hold for a limbo target and for a crash cash-out point alike. Each multiplier times its chance comes to 0.99 of the stake, so the edge is the same on every row; what changes down the table is only how rarely the bet wins.

A mines board hides a chosen number of mines among its squares. Under the usual published rule the mines are placed before the first click, one random number per mine, each picking a position among the squares still free so that no square is chosen twice. A click reveals either a gem, which raises the multiplier, or a mine, which loses the stake; the player can cash out after any safe square.

The chance of clearing several squares in a row is a product of shrinking fractions: safe squares left over squares left, once per click. With 3 mines, opening 5 safe squares in a row succeeds 49.57% of the time and pays 2.00x. The multiplier paid is 0.99 divided by that chance, so the edge is identical however many squares are opened and whenever the player stops; more mines make each click riskier and each step richer, but not cheaper.

Multipliers follow the common published form, 0.99 divided by the chance, and are rounded to two decimals. A casino that pays a different ladder prints it in the game, and dividing its multiplier into 0.99 shows the chance it has priced in.

Each row of pins takes one random number in the common published form: below one half the ball drops left, otherwise right, and the slot it lands in is the count of right turns. The landing slot is therefore a binomial draw, the same spread of results as counting heads in a run of coin tosses, heavy in the middle and thin at the edges.

On a 16-row board the ball reaches an outermost slot only when every turn goes the same way, one drop in 65,536 for each edge. The five slots in the middle catch 79.0% of all drops, and the single center slot 19.64%.

The prizes printed under the slots differ from one casino to another, and within one game by risk setting and row count, so a slot's chance can be given here but its prize only on the game itself. The edge follows from the two together: multiply each slot's prize by its chance and add the results. Plinko titles from game studios, each with its own board and prize table, are collected on the plinko pages.

Every rule on this page is priced the same way: the payout times the chance of earning it comes to a fixed share of the stake, whichever target is picked. The expected result of a single bet is therefore a loss of the edge times the amount staked, at a low target as at a high one, with few mines as with many, on a cautious cash-out as on a greedy one.

Betting systems run into the same arithmetic. Doubling after a loss, raising the target after a win or switching games mid-session changes the size and order of the bets, not their price, so the average loss over a session is the edge multiplied by the total amount wagered. A plan that wagers more in total loses more on average, however its bets are arranged. What a target does change is the ride: a high target wins rarely and pays a lot, a low one wins often and pays little, and a balance swings harder the higher the target. Play only with money you can afford to lose.

The rule: Over many bets, the average loss is the edge times the total staked. Target, cash-out point and bet sizing decide how bumpy the ride is, not what it costs.