Expected value and variance in dice

Take a stake of 1 unit at win chance p (as a fraction) and payout multiplier m. A win returns m, so the profit is m − 1. A loss costs the stake, −1. Weighting each outcome by its probability:

With the dice payout m = (100 − edge) / chance and p = chance / 100, the product p · m is (100 − edge) / 100. So EV is −edge / 100 per unit, the same at every chance. At a 1% edge every bet loses 0.01 units on average. The house edge guide covers this in more detail.

Variance is the expected squared distance from the mean. For a two-outcome bet whose results differ by m (a win at m − 1, a loss at −1), it reduces to a short formula. The standard deviation σ is its square root and is in the same units as the stake.

Both chances have the same EV. The 3.35% bet has about 28.3 times the variance of the 49.5% bet, because it loses most of the time and pays 29.5522x when it wins. One unit staked at 3.35% swings by about 5.32 units per bet, compared with about 1 at 49.5%.

Dice rolls are independent, so over n flat bets of 1 unit the means add and the variances add. The expected profit is n · EV and its standard deviation is σ · √n.

The expected loss grows in proportion to n, while the spread grows only with its square root. Early on the spread dominates. Later the drift does. A useful marker is the number of rolls at which the expected loss equals one standard deviation:

At 49.5% that is about 9,999 rolls. At 3.35% it is about 282,766 rolls. Before those points, a run that is one standard deviation lucky still shows a profit.

The table below gives the exact probability that a flat bettor is strictly ahead after n bets. It is computed from the binomial distribution: with k wins the profit is k · m − n, so the run is ahead when k > n / m.

After 10,000 bets at 49.5%, about 15.6% of players are still ahead. After 100,000 it is 0.077%. At 3.35% the variance is so large that 42.0% are ahead after 10,000 bets and 27.7% after 100,000.

At 3.35% the share rises between 100 and 1,000 rolls before it falls. With few bets, the number of wins needed to get ahead moves in whole steps, and at 3.35% one extra win is worth about 29.6 units. The trend is still clear: the longer the run, the smaller the share of players who are ahead.

This is why a strategy that made a profit over a few thousand rolls says little about its quality. Many players with a strategy that has the same negative EV will report the same thing over the same number of rolls.

The formulas above are for flat betting. A progression changes the bet size from roll to roll, so the variance of a session depends on the bets it places. EV still adds up bet by bet: the session's expected result is −edge / 100 times the total wagered.

What changes is the shape of the outcome. A martingale wins small amounts in most sessions and loses a large amount in a few, which gives a distribution with a long left tail. The median session can be positive while the mean is negative. The risk of ruin guide computes how often the large loss arrives.

The Stress Test tab in the simulator runs the current script on 5 workers with different random seeds and the same code and settings. The overlaid balance lines show the spread directly, and the summary lists the best, worst, mean and median profit and how many runs went bankrupt.

Try a flat script at 49.5% for 10,000 rolls, then the same at 3.35%. The 3.35% lines fan out much further, which is the variance table above in chart form. The guide on reading simulation results explains each statistic.

All guides are listed on the Learn hub. Other articles on dice math:

• Risk of Ruin Explained: Formula for Dice Betting
• Dice House Edge Explained: Payout & Expected Loss
• Dice Bankroll Management: Bet Sizing & Stop Loss
• How to Read a Dice Simulation: Stats Explained