Learn / Risk of Ruin
Risk of ruin is the chance of losing the money set aside for play before you stop.
It is always stated for a bankroll, a betting rule, a game and a horizon. Change any of
them and the figure changes.
Horizon
Ruin within a stated period is not the same as ruin at any time.
Finite horizon
Within a Stated Period
The chance of ruin within a stated number of rounds, hours or sessions — or before a stated target is reached.
Unlimited horizon
At Any Time
The chance of ruin at some point if play continues indefinitely under the same game and the same betting rule.
The finite-horizon risk is never higher.
For the same game, betting rule and bankroll, the risk of ruin within a stated period
is never higher than the risk on an unlimited horizon: being ruined within that period
is one of the ways of being ruined at some point.
Certain ruin holds in a stated model, not for every game.
In the classic gambler’s-ruin model — independent bets that win or lose
exactly one stake, constant probabilities, a fixed stake, a finite bankroll, no
top-ups and no stopping target — the unlimited-horizon risk of ruin is 1 (100%)
whenever the win probability p is 0.5 or less, however large the bankroll. This is a
result of that model. It is not presented as a universal theorem for every game with
a zero or negative EV.
An illustrative example
The same small edge, with different bankrolls and bets.
Assumptions of this example.
An illustrative game, not blackjack and not OOB: each round is an independent,
even-money bet; the player wins with probability 50.5% and loses with probability
49.5%; there are no pushes, doubles, splits or blackjacks; the bet is fixed; the
bankroll is finite and is not topped up; play continues without a stopping target.
This is the classic gambler’s-ruin model. The figures describe this game only. They are
not estimates for OOB, for blackjack or for any player.
| Bankroll |
Bet per round |
Bankroll in bets |
Unlimited-horizon risk of ruin |
| 50 units |
1 unit |
50 |
≈ 36.8% |
| 100 units |
1 unit |
100 |
≈ 13.5% |
| 200 units |
1 unit |
200 |
≈ 1.8% |
| 100 units |
2 units |
50 |
≈ 36.8% |
Illustrative game only. Computed with the classic gambler’s-ruin result for this game: risk = (0.495 ÷ 0.505) raised to the number of bets in the bankroll, rounded.
100 units, 1-unit bets
→
at least 100 rounds are needed to lose 100 units
Within 99 rounds
→
risk of ruin = 0
Unlimited horizon
→
risk of ruin ≈ 13.5%
Illustrative game only. Longer finite horizons give values between these two; a stopping target, such as ending play after a stated gain, also lowers the figure.
What drives the risk
Bankroll, bet size, edge and dispersion.
Bankroll
Bets the Bankroll Can Absorb
The more bets the bankroll can absorb, the lower the risk. In the illustrative game, going from 50 to 200 bets lowers it from about 36.8% to about 1.8%.
Bet size
A Larger Bet Shrinks the Bankroll
In the classic model, raising the bet with the same bankroll has the same effect as shrinking the bankroll: 100 units with 2-unit bets behaves like 50 units with 1-unit bets.
Edge
In the Classic Model, No Edge Means Ruin
In the classic ±1 model, the unlimited-horizon risk is below 100% only when p is above 0.5. Within that model, the smaller the edge, the larger the bankroll needed for the same risk.
Dispersion
Variance Within an Approximation
Under the diffusion approximation below, the same EV with a larger variance gives a higher risk. EV and variance alone do not determine the exact risk for an arbitrary distribution of round results. See
Variance.
Diffusion (Brownian) model
→
risk ≈ e−2 × EV × bankroll ÷ variance (constant positive drift, ruin at zero; EV and variance per round, in bets)
EV +0.01, 100 bets, variance 0.9999
→
e−2.0002 ≈ 13.5% (the exact classic-model value is also ≈ 13.5%)
EV +0.01, 100 bets, variance 1.3
→
e−1.5385 ≈ 21.5%
Illustrative values only. The formula is exact for a Brownian model with a constant positive drift and ruin at zero; for discrete games, such as the illustrative one, it is an approximation, not a general exact formula. The variance of 1.3 is an assumed value chosen to show the direction of the effect within this approximation; it is not a measured figure for blackjack or for OOB.
Doubles and splits
In blackjack, the money committed in a round can exceed the initial bet.
A full double adds an amount equal to the original bet; a split adds a bet equal to the
original on the new hand, and further splits add more. They also widen the spread of
round results.
Illustrative: 3 units left
→
an initial bet of 2 units can be placed (2 ≤ 3)
A double or a split is called for
→
2 + 2 = 4 units are needed, more than the 3 left
Consequence
→
the next commitment cannot be covered; the action cannot be taken as intended
Under the practical definition above, this already counts as ruin. A risk-of-ruin figure
for blackjack therefore depends on whether it allows for the money that doubles and
splits can require.
In the application
OOB does not yet provide risk-of-ruin estimates.
The suggested bet has no risk-of-ruin estimate.
The suggested initial bet uses fixed levels of 0, 1, 2 or 13 units, chosen from the
pre-bet EV estimate. A unit is your confirmed balance divided by 100 and rounded to the
nearest whole number (halves up). The bet is fixed for the round: a double or a split
uses the same initial stake, and a balance confirmed during the round does not change
it. The same levels are labelled OOB Selective in the
100-Hour Simulated Replay.
OOB does not calculate a risk of ruin for this rule, and no figure on this page
describes it. At the highest level, the initial bet alone is about 13% of the
confirmed balance; because of the rounding it is not always exactly 13%. The relative
effect of rounding can be greater at smaller balances. A full double or a split adds
the same amount again.
Example: confirmed balance 2,550
→
unit = 2,550 ÷ 100 = 25.5, rounded to 26
Highest level, 13 units
→
13 × 26 = 338, fixed for the round
Share of the balance
→
338 ÷ 2,550 ≈ 13.25%
Illustrative arithmetic of the published rule, not a risk estimate.
What the suggested bet does
It changes the initial stake with the pre-bet EV estimate and with the balance you
confirm, and keeps it fixed for the round.
What it does not do
It does not remove the risk of losing the bankroll, and it cannot guarantee a
profit. Its risk of ruin has not been estimated.
The rule is described under bankroll, bet sizing and risk in the
Methodology.