Do dice betting systems work?

The expected value of a bet is its average result over a very large number of identical bets. At a 1% house edge, a bet of 1 unit has an expected value of −0.01 units at any win chance. A lower chance pays more when it wins and wins less often, and the two balance out at the same figure:

Expected values add up. A session’s expected result is the sum over its bets, which comes to −1% of the total amount wagered. A system can make the total wagered larger or smaller, and it can make some bets big and others small, but every unit staked carries the same −1%.

What about choosing bet sizes, or when to stop, based on earlier results? That is the part most systems rely on, and it cannot help either. The next roll does not depend on the previous ones, so the information a system uses to size the next bet says nothing about whether that bet will win. A stopping rule such as “quit once I am 10 units up” changes how often a session ends ahead. It pays for that with rarer, larger losing sessions, and the average across all sessions stays at −1% of everything wagered. Probability theory has a name for this result (the optional stopping theorem), but the intuition above is all it says for dice.

The gambler’s fallacy is the feeling that a win is due after a run of losses. In provably fair dice each roll comes from an HMAC of the seeds and a nonce that goes up by one per bet. The previous results are not part of the calculation, so ten losses in a row leave the chance of the next roll winning exactly where it was. The provably fair guide shows the calculation step by step.

Long streaks are also more common than intuition suggests. At 49.5% chance, a run of 10 losses starts on any given roll with probability 0.108%, and on average one shows up every 1,871 rolls. A bot placing several bets a second meets such a streak many times a day. Seeing one is ordinary, and it does not make a win any more likely afterwards.

Martingale doubles the stake after each loss and goes back to one unit after a win. At 49.5% chance and a 1% edge the payout is 2.00x, so the first win in a cycle recovers every earlier loss in that cycle and adds exactly 1 unit. Win rates are very high, which is why martingale looks like it works for a while.

Now give it a bankroll of 1,023 units, enough for 10 bets in a row: 1, 2, 4 and so on up to 512. A cycle fails only when all 10 lose.

The average of −0.1046 units per cycle is exactly the house edge at work. An average cycle stakes 10.46 units in total, because the larger bets are only placed after earlier losses, and 1% of that is 0.1046. The doubling left the rate of loss where it was and moved the loss into a single rare event.

That event arrives about once every 927 cycles. On average the bettor wins about 926 units, one cycle at a time, before losing 1,023 at once. The chance of hitting the fatal streak grows with play:

A larger bankroll pushes the bust further out but never removes it, and the amount lost when it comes grows at the same rate as the bankroll. The risk of ruin guide turns this into a formula, and the calculator runs it for any base bet and bankroll.

Casinos set a maximum stake per bet, and it caps a doubling system whatever the bankroll. With a 1-unit base bet and a max bet of 100 units, the progression can place 7 bets (1, 2, 4 and so on up to 64). The next doubling would be 128 units, which the casino refuses.

At 49.5% that cap is reached once in about 119 cycles, far more often than the 10-bet example above, and more bankroll does not help. Raising the base bet makes it worse, because the limit is reached in fewer doublings. Casino limits differ, so check the maximum stake where you play before relying on any progression.

Positive progressions raise the stake after wins instead of losses. Paroli is the common example: let the winnings ride for three wins in a row, then bank the run and start over. At 49.5% chance and a 2.00x payout, each cycle has two possible results:

• Three wins in a row, probability 12.13%: a profit of 7 units.
• Any loss before that, probability 87.87%: a loss of 1 unit, because every larger stake was made of earlier winnings.

The average per cycle is −0.0297 units. An average cycle stakes 2.9701 units, and 1% of that is again the same figure. Paroli and martingale sit at opposite ends: one has frequent small losses and rare larger wins, the other frequent small wins and rare large losses. Both lose 1% of what they wager. The Paroli script runs this system in the simulator.

These systems differ in how fast the stake grows after a loss. D’Alembert adds one unit per loss, Fibonacci moves along the sequence 1, 1, 2, 3, 5, 8 and so on, Labouchere bets the sum of the first and last numbers on a list, and Oscar’s grind raises the stake only after wins while chasing a one-unit profit per series. Slower growth means a bankroll survives longer losing runs, and each recovery takes longer. None of them changes the −1% per unit wagered.

A system is a choice about the shape of results, and that choice is real even though the average is fixed.

• Variance: progressions stake more after some outcomes than others, which widens or narrows the spread of session results.
• Hit rate: martingale finishes most cycles ahead and Paroli finishes most cycles behind, at the same win chance per roll.
• Risk of ruin: fast progressions concentrate losses in rare streaks that can take the whole bankroll. See risk of ruin explained.
• Total wagered: bigger average stakes mean a bigger expected loss, since the loss is a fixed share of turnover. See the house edge guide.