Strategy

Low chance dice strategy (1% to 5%)

A low chance dice strategy bets at a win chance of about 1% to 5% and waits for payouts between roughly 20x and 99x. Wins are rare, so the strategy lives or dies on how it handles the long losing runs in between. This page gives the payouts, how long those dry spells last, the smallest loss multiplier that still recovers, and how a preroll changes the bankroll a strategy needs.

All figures assume a 1% house edge and are computed with the same formulas as the DiceSim calculator.

Payouts at 1%, 2%, 3.35% and 5%

The payout multiplier is (100 - house edge) / win chance. The average gap between wins is 100 divided by the chance, and the median wait is the number of rolls after which there is a 50% chance of having won at least once.

Low chance dice payouts and waiting times at a 1% house edge
Win chance	Payout	Average gap	Median wait	Spread per 1-unit bet
1%	99.00x	100.0 rolls	69 rolls	9.85
2%	49.50x	50.0 rolls	35 rolls	6.93
3.35%	29.55x	29.9 rolls	21 rolls	5.32
5%	19.80x	20.0 rolls	14 rolls	4.32

The last column is the standard deviation of the result of a 1-unit bet. It is about 1.00 at 49.5% and nearly 10 at 1%, so the same stake swings much harder at low chance. The expected result per unit is −0.01 in every row. The 1% odds, 3.35% odds and 5% odds pages go through each chance in more detail.

Why dry spells are long

At 3.35% chance the average gap between wins is about 30 rolls, but averages hide the tail. The probability that a given roll starts a run of N losses is (1 - chance/100)^N, and the expected number of rolls between two such runs is (1 - q^N) / ((1 - q) × q^N), where q is the loss probability. The table shows dry spells of three, five and eight times the average gap:

Expected rolls between losing streaks at low win chances
Win chance	Streak (losses)	Chance a roll starts it	Expected rolls between
1% (3x gap)	300	4.904%	1,939
1% (5x gap)	500	0.657%	15,120
1% (8x gap)	800	0.032%	310,245
2% (3x gap)	150	4.830%	985
2% (5x gap)	250	0.640%	7,756
2% (8x gap)	400	0.031%	161,587
3.35% (3x gap)	90	4.658%	611
3.35% (5x gap)	149	0.624%	4,755
3.35% (8x gap)	239	0.029%	102,703
5% (3x gap)	60	4.607%	414
5% (5x gap)	100	0.592%	3,358
5% (8x gap)	160	0.027%	73,305

A dry spell of five times the average gap turns up about once every 3,358 to 15,120 rolls, depending on the chance. At three bets per second that is between about 19 and 84 minutes of play. At 3.35%, a run of 239 losses comes about once every 102,703 rolls, which is rare per roll and routine over a long session.

Flat betting at low chance

With a flat stake, a losing run of N costs exactly N units, so the bankroll needed is easy to state. To cover the losing run that starts on only 1 roll in 1,000 you need:

Flat betting bankroll for the 1-in-1,000 losing streak
Win chance	Streak with 0.1% chance per roll	Bankroll (units)	Expected rolls between
1%	688	688	100,590
2%	342	342	50,029
3.35%	203	203	30,099
5%	135	135	20,320
49.5%	11	11	3,706

To survive a losing run of the same 1-in-1,000 rarity, a flat bettor needs 688 units at 1% and 11 at 49.5%. Flat betting at low chance is still cheap compared with any progression, because the stake never grows.

The minimum loss multiplier that recovers

Many low chance scripts raise the stake by a small factor r after each loss so that the eventual win covers the losses before it. A win at a payout of M returns M - 1 times its stake as profit. For that profit to cover every earlier loss in the run, at any length of run, the factor has to be at least:

textMinimum recovery multiplier

r_min = M / (M - 1)

At exactly r_min, a win at any point leaves the run M - 1 units ahead, the same as a win on the first roll. A smaller factor recovers on short runs and falls behind on long ones. At 49.5% the payout is 2.00x and r_min is 2, which is plain martingale. At low chance the minimum is small:

Minimum recovering loss multiplier and the bankroll it needs for the 1-in-1,000 streak
Win chance	Payout	r_min	Increase per loss	Bankroll at r_min (units)
1%	99.00x	1.0102	1.02%	105,754
2%	49.50x	1.0206	2.06%	52,076
3.35%	29.55x	1.0350	3.50%	30,912
5%	19.80x	1.0532	5.32%	20,523
49.5%	2.00x	2.0000	100.00%	2,047

The last column is the cost of surviving the same 1-in-1,000 streak from the flat-betting table while growing the stake at r_min. It is tens of thousands of units at every low chance, against 2,047 for martingale at 49.5%. Small per-loss increases compound over hundreds of losses.

Preroll and delayed escalation

A preroll is a number of losses a script waits through at the minimum stake before it starts multiplying. At 3.35% half of all runs end in a win within 21 rolls, so the preroll keeps those cycles cheap and saves the escalation for the long droughts.

DiceSim’s delayed escalation script defaults to chance = 3.35 and basebet = balance / 50000. It counts losses in a row in lossRun. While lossRun is 50 or less, the next bet is the base bet. Beyond that, each loss multiplies the previous stake by 1.04. A win resets the count and the stake, and after 240 losses in a row the script prints the drawdown and calls stop().

luadelayed-escalation.lua (settings and dobet)

local preroll    = params.preroll or 50        -- losses to wait before escalating
local multiplier = params.multiplier or 1.04   -- growth per loss once escalating
local maxLosses  = params.maxlosses or 240     -- bail out (stop) if the drought is absurd

local lossRun = 0

function dobet()
  if win then
    lossRun = 0
    nextbet = basebet
    return
  end

  lossRun = lossRun + 1
  if lossRun >= maxLosses then
    print("Stopping: " .. lossRun .. " losses in a row, drawdown " .. string.format("%.8f", -profit))
    stop()
    return
  end

  if lossRun > preroll then
    nextbet = previousbet * multiplier
  else
    nextbet = basebet
  end
end

What the full drought costs

In one losing run the first 51 bets are 1 unit each, and from bet 52 each stake is 1.04 times the one before. The 240 bets of a run that never hits add up to 43,105 units, and the last of them is 1,657 units. That fits inside the default bankroll of 50,000 units. A run of 240 losses at 3.35% starts on a given roll with probability 0.028%, or about once every 106,264 rolls.

The hits that still lose money

The 1.04 growth rate is only a little above r_min = 1.0350 for this payout, and the preroll losses are not covered by it. A win on the first 29 bets of a run makes a profit. A win anywhere from bet 30 to bet 100 ends the run behind, because the flat losses have grown past the 28.55 units a 1-unit win returns and the escalation needs time to catch up. From bet 101 onwards, a hit is profitable again.

That gap is the price of the cheap preroll. A longer preroll widens it, and a faster growth rate narrows it while making the full drought far more expensive. The simulator lets you change params.preroll, params.multiplier and params.maxlosses and see the effect on drawdown directly.

Risk compared with 49.5%

Low chance and 49.5% betting lose the same 1% of every unit wagered on average. They differ in how the result is distributed.

Each 1-unit bet at 1% has a standard deviation of 9.85 units against 1.00 at 49.5%, so session results spread much wider.
Flat bets at low chance are cheap and slow: a long dry spell costs units one at a time, and a single hit can wipe out hundreds of rolls of losses.
Recovery progressions at low chance grow slowly per loss but run for hundreds of losses, so the cost of the rare full drought is large next to the base bet. A martingale at 49.5% reaches its limit after about ten losses, which happens far more often.

Neither profile beats the house edge. The choice is between frequent small swings and rare large ones, and the risk of ruin guide shows how to put a number on the second.

Practical points

Real stakes have a smallest increment, such as one satoshi for bitcoin, so a multiplier such as 1.0102 may not change a very small stake at all. Check that your base bet is large enough for the growth to show up.
Size the bankroll from the longest run you intend to survive, not the average gap. The dice calculator gives the survivable streak for any chance, base bet and multiplier.
Decide on a stop rule before you start. The template’s maxlosses setting is one; a loss limit on profit is another.
Simulate before you bet. One run at low chance is one draw from a very wide distribution, so use the stress test to compare several seed pairs.

FAQ

What is the payout at 1% chance?

At a 1% house edge the payout multiplier is (100 - 1) / 1 = 99.00x. At 3.35% it is 29.55x and at 5% it is 19.80x.

How long can a dry spell at 1% chance last?

The average gap between wins is 100 rolls, but about half the time the first win takes 69 rolls or more, and a run of 688 losses starts on roughly 1 roll in 1,000.

Is low chance dice less risky than 50/50?

The expected loss per unit wagered is the same at every chance. Low chance bets swing much more per bet, so flat betting there needs more units of bankroll to ride out the same level of bad luck.

What is a preroll in dice?

A number of losses a strategy waits through at the minimum stake before it starts raising bets. DiceSim's delayed escalation script waits 50 losses at 3.35% and then multiplies the stake by 1.04 per loss.

What loss multiplier recovers all losses at low chance?

A progression recovers everything on any win when it multiplies the stake by at least M / (M - 1) per loss, where M is the payout. That is 1.0102 at 1% and 1.0350 at 3.35%, or about 1.02% and 3.50% more per loss.