Standard deviation and risk of ruin

A single hand of basic-strategy blackjack has a standard deviation of about 1.14 betting units against an expected loss of about 0.004 units, so the swing on any one hand is nearly three hundred times its expected cost. Over many hands the expectation grows in proportion to the number of hands while the spread grows only with its square root, and that single relationship explains both why short sessions are unpredictable and why the house edge always wins in the end.

Where 1.14 comes from

The Wizard of Odds variance page reports a simulation of about ten billion hands under liberal six-deck S17 rules (double any two, double after split, late surrender, resplit aces, three resplits): variance 1.303 and standard deviation 1.142 per hand, for a house edge of 0.28%. A simple even-money bet would have a standard deviation of exactly 1. Blackjack’s is higher because:

• doubles and split hands risk two or more units;
• naturals pay 1.5 units;
• surrendered hands lose 0.5 units.

The same page gives a variance of 1.295 for the Wong benchmark rules (six decks, S17, no DAS, no surrender, no resplit aces), whose square root is 1.138. The Wizard’s house-edge comparison table rounds blackjack to 1.15. Any value from 1.14 to 1.15 is defensible for the site’s profile; the arithmetic below uses 1.142 and a house edge of 0.41% (Wizard of Odds, six decks, S17, DAS, no surrender).

Expectation and spread over N hands

Treat each hand as independent with mean μ = −0.0041 units and standard deviation σ = 1.142 units. Over N hands:

Expected result = N × μ

Standard deviation = σ × √N

| Hands (N) | Expected result (units) | Standard deviation (units) | SD ÷ |expectation| | | ---: | ---: | ---: | ---: | | 1 | −0.004 | 1.14 | 279 | | 100 | −0.41 | 11.42 | 28 | | 400 | −1.64 | 22.84 | 14 | | 1,000 | −4.10 | 36.11 | 8.8 | | 10,000 | −41.0 | 114.2 | 2.8 |

Setting σ√N = N|μ| and solving gives N = (σ/μ)² = (1.142 / 0.0041)² ≈ 77,600 hands before the expected loss equals one standard deviation. At 100 hands an hour that is about 780 hours of play. Until then the result of your play is dominated by variance; after it, the house edge becomes the visible trend.

A worked example in pounds

Suppose £10 flat bets, 400 hands (a long evening), the same model:

Expected result: 400 × £10 × (−0.0041) = −£16.40.

Standard deviation: £10 × 1.142 × √400 = £10 × 1.142 × 20 = £228.40.

Under a normal approximation about two-thirds of such sessions finish between −£245 and +£212 (expectation ± one SD) and about 95% between −£473 and +£440. Finishing £200 ahead is unremarkable; so is finishing £200 behind. Neither tells you anything about whether you played well. The bankroll and variance guide applies the same reasoning to budgeting.

The normal approximation is adequate for hundreds of hands. For a handful of hands the actual distribution is lumpy (most results are exactly −1, +1, 0, ±2 or +1.5) and the approximation should not be trusted for tail probabilities.

Risk of ruin: the formula

Risk of ruin is the probability of losing a bankroll of B units before some stopping point. Wikipedia’s risk-of-ruin page gives an approximation for a random walk with per-step mean μ and standard deviation σ starting at s units:

P(ruin) = (2 / (1 + μ/r) − 1)^(s/r), where r = √(μ² + σ²).

For blackjack, μ is tiny relative to σ, and the expression collapses to the familiar shorthand P(ruin) ≈ exp(−2μB/σ²). The two forms agree to four decimal places for every row below. Both apply only when μ is positive; with μ ≤ 0 and unlimited play, ruin is certain.

Risk-of-ruin model outputs

The table below is a model output from the Wikipedia formula with σ = 1.142, not a measured frequency. It applies to a player with a genuine positive edge, which for a flat-betting basic-strategy player does not exist; the edges shown are what a skilled card counter might achieve on average across a bet spread, and even then the counter’s variance is higher than 1.303 because bets are not flat. Treat the figures as illustrative.

Two things stand out. Risk of ruin falls exponentially with bankroll, so doubling the bankroll squares the (fractional) risk: 21.6% becomes 4.7% at 400 units and 2.2% at 500. And it is acutely sensitive to edge: at +0.5% a 200-unit bankroll fails one time in five, at +1.5% one time in a hundred. A counter who overestimates their edge by half a percent is working from a wildly wrong risk figure, which is the practical warning in the bet spreads and bankroll article.

The basic-strategy player: ruin is a matter of time

For the ordinary player μ is negative and the formula gives 100%. The useful question becomes: how likely is a bankroll of B units to be lost within N hands? A normal-model estimate (again illustrative, μ = −0.0041, σ = 1.142):

These are computed as twice the probability of finishing below −B (the reflection principle, which approximates the chance of touching −B at any point). A bankroll of 100 units at a fixed stake is very unlikely to be exhausted in an evening; one of 25 units will be gone in about half of thousand-hand stretches. None of this changes the expectation: a bigger bankroll buys time, not a better game, a point the betting systems article makes about progression schemes that claim otherwise.

Related reading

• How the house edge is calculated
• Bankroll and variance
• Bet spreads and bankroll for counters
• Betting systems debunked