Variance — OddsOfBlackjack

Five terms

Average, outcome, spread, typical deviation and the fall from a peak.

They answer different questions and are easy to confuse. None of them, on its own, describes what the next session will bring.

The average

Expected Value (EV)

The probability-weighted average result of a bet or a decision. It describes the average over many repetitions, not the next result. See Expected Value.

The outcome

Realised Result

What actually happened over the rounds played. It is one outcome among the many that were possible.

The spread

Variance

The probability-weighted average of the squared differences between the possible results and the EV. It is measured in squared units.

The typical deviation

Standard Deviation (SD)

The square root of the variance, in the same units as the result. It gives the typical size of a deviation from the EV.

The fall from a peak

Drawdown

A fall of the result of play from a previous peak to a later low, with deposits and withdrawals kept separate. The maximum drawdown is the largest such fall over a period.

An illustrative game

A small positive edge, and how far results can stray from it.

Assumptions of this example.

An illustrative game, not blackjack and not OOB: each round is an independent, even-money bet of 1 unit; the player wins with probability 50.5% and loses with probability 49.5%; there are no pushes, doubles, splits or blackjacks. The figures describe this game only. They are not estimates for OOB, for blackjack or for any player.

EV per round → 0.505 × (+1) + 0.495 × (−1) = +0.01 unit

Variance per round → 1 − 0.01² = 0.9999 (each result is +1 or −1, so its square is 1)

SD per round → √0.9999 ≈ 1 unit

Over n independent rounds of this game, same stake → EV = 0.01 × n; variance = 0.9999 × n; SD ≈ √n

Rounds	EV of the total	SD of the total	Chance of being behind
100	+1 unit	≈ 10 units	≈ 42%
1,000	+10 units	≈ 31.6 units	≈ 36%
10,000	+100 units	≈ 100 units	≈ 16%

Illustrative game only. “Chance of being behind” is the probability that the total result is below zero after that many rounds, computed exactly from the binomial distribution of wins and rounded. With an even number of rounds the total can also end exactly at zero; that case is not counted as behind. In this game — independent rounds with the same distribution and the same stake — the EV grows in proportion to the number of rounds and the SD only with its square root. With changing stakes or dependent rounds, these conclusions do not carry over automatically.

Why positive EV can still lose

Over a short run, the spread is much larger than the edge.

After 100 rounds of the illustrative game, the EV of the total is +1 unit while its SD is about 10 units, so being behind (about 42%) is nearly as likely as being ahead (about 50%). Even after 10,000 rounds, the EV of +100 units is only about one SD above zero.

What a positive EV means

In the illustrative game, the average result per round is above zero, and over many more rounds the share of outcomes that end behind tends to fall.

What it does not mean

It does not mean that a hand, a session or any finite run ends ahead, and it does not set a limit on how deep a drawdown can be.

Illustrative balances → 1,000 → 1,150 → 980 → 1,060 → 1,200 units, with no deposits or withdrawals (not a measured result)

Peak, then low → 1,150, then 980

Drawdown → 1,150 − 980 = 170 units (170 ÷ 1,150 ≈ 14.8% of the peak)

Illustrative sequence only. It ends at a new high of 1,200 units, yet it contains a drawdown of 170 units. Ending ahead and going through a drawdown are not exclusive.

Deposits and withdrawals are not results.

A drawdown is measured on the result of play. Money added to or taken from the bankroll must be kept separate: topping up after a loss raises the balance, but it is not a recovery of that loss, and a withdrawal is not a loss of play.

What changes the spread

Stake, bet changes, the shape of the round and the number of rounds.

Bet size

Scale by k, Variance by k²

If only the stake changes and the distribution of the result per unit staked stays the same, multiplying the stake by k multiplies the EV and the SD by k, and the variance by k². In the illustrative game, 2-unit bets give an EV of +0.02 and an SD of about 2 units per round.

Changing bets

Stake Changes the Spread

Holding the per-unit result distribution fixed, a round’s variance scales with the square of its stake. The spread of the total also depends on any dependence between rounds. The √n pattern of the illustrative game then no longer applies directly.

Shape of the round

Doubles, Splits, Blackjacks

In blackjack, doubles, splits and the blackjack payout give a round more possible results than a plain even-money bet, which widens the spread of round results.

Number of rounds · illustrative game

Smaller Share Behind, Never Zero

In the illustrative game — independent rounds, the same distribution and the same stake — the SD of the total grows with √n and its EV with n, so over many more rounds the share of outcomes that end behind tends to fall, without reaching zero after any finite number of rounds. For changing stakes or dependent rounds, this is not carried over automatically.

In the application

OOB does not yet provide these estimates.

Explained here, not calculated in the application.

OOB shows the EV of each available action and suggests an initial bet. It does not yet calculate variance, standard-deviation or risk-of-ruin estimates for your play. The figures on this page are illustrative arithmetic, not outputs of the application.

The One Million Cards and 100-Hour Simulated Replay pages report realised drawdowns on simulated tracks. They are realised results, not estimates of variance and not forecasts.

Continue

Related pages and published evidence.

Risk of Ruin

Running out of bankroll, on a finite or an unlimited horizon.

Read the Page →

Expected Value

Action EV, the pre-bet estimate and the realised result.

Read the Page →

100-Hour Simulated Replay

Realised results and drawdowns on one simulated window.

See the Replay →