The Mathematics of Volatility in Video Poker

Volatility (or variance) measures how much results deviate from expected value over time. Understanding volatility mathematics is essential for bankroll management, game selection, and realistic expectations in video poker play.

Defining Volatility

The Statistical Foundation

Volatility is measured by variance and standard deviation:

Variance (σ²): Average squared deviation from mean

Standard Deviation (σ): Square root of variance

For video poker:

$$\sigma = \sqrt{\sum_{i} p_i \times (x_i - \mu)^2}$$

Where:

p_i = Probability of outcome i

x_i = Payout for outcome i

μ = Expected value (mean return)

Volatility Metrics

Per-Hand Standard Deviation

GameStandard DeviationClassification9/6 Jacks or Better4.42Low9/6 Double Double Bonus6.44Medium-HighFull Pay Deuces Wild5.08MediumNot-So-Ugly Deuces4.03Low10/7 Double Bonus5.47MediumSuper Double Double Bonus7.15High

What These Numbers Mean

Standard deviation represents typical deviation per hand:

4.42 for 9/6 JoB: Results typically vary by about 4.4 betting units from expected

6.44 for 9/6 DDB: More extreme swings—about 50% more volatile

Variance Components

What Creates Volatility

Volatility comes from paytable structure:

Low Volatility Factors:

Frequent small payouts

Smaller jackpot premium

Balanced payout distribution

High Volatility Factors:

Infrequent but large payouts

High jackpot multipliers

Reduced low-tier payouts

The Two Pair Effect

Comparing Jacks or Better to Double Double Bonus:

HandJoB PayoutDDB PayoutTwo Pair21Full House99Four Aces w/kicker25400

DDB reduces Two Pair (frequent) to fund Four Aces bonus (rare), dramatically increasing variance.

Bankroll Requirements

The Risk of Ruin Concept

Risk of Ruin (RoR): Probability of losing entire bankroll before achieving goal.

Factors affecting RoR:

Bankroll size

Game volatility

Bet size

Target duration

Bankroll Guidelines

Conservative estimates for 1% Risk of Ruin:

GameSDBankroll (5-coin bets)9/6 JoB4.423,000 units9/6 DDB6.446,500 units10/7 DB5.474,500 units

The Royal Flush Factor

Much of video poker variance comes from the Royal Flush:

Probability: ~1 in 40,000 hands

Contribution to RTP: 2-3%

Effect: Long periods "below expectation"

Until you hit a Royal, you're playing a losing game even on positive EV machines.

Session Volatility

Short-Term Expectations

For a 1,000-hand session on 9/6 Jacks or Better:

Expected loss: ~$2.30 per $500 wagered (0.46% edge)

Standard deviation for session:

$$\sigma_{session} = \sigma_{hand} \times \sqrt{n} = 4.42 \times \sqrt{1000} = 139.8 \text{ units}$$

Practical meaning:

~68% of sessions fall within ±139.8 units of expectation

~95% fall within ±279.6 units

~99.7% fall within ±419.4 units

Winning Session Probability

Even on negative EV games, winning sessions are common:

Session LengthProbability of Winning100 hands~48%500 hands~46%1,000 hands~44%5,000 hands~38%

Short-term variance overwhelms small house edge.

Multi-Hand Volatility

How Multi-Hand Affects Variance

Playing multiple hands changes volatility profile:

Total session variance:

$$\sigma_{multi} = \sigma_{single} \times \sqrt{hands}$$

Per-deal variance:

Increases with number of hands

Triple Play vs. Single Hand

Same initial hand dealt across multiple lines:

MetricSingleTripleBet per deal5 coins15 coinsResults per deal13Per-deal varianceStandard~1.73×Session varianceStandard~1.73× for same deals

The 100-Play Extreme

100-hand games create massive volatility:

Per-deal standard deviation: 10× single hand

Royal Flush across all hands: Life-changing

Complete miss on good draw: Devastating

Not for underfunded players

Game Selection by Volatility

Choosing Based on Goals

Low Volatility (Jacks or Better, NSUD):

Longer play sessions

Smaller bankroll requirement

Entertainment-focused

Lower emotional stress

High Volatility (DDB, Super DDB):

Jackpot chasing

Larger bankroll required

Thrill-seeking

Tolerance for losses

The Recreational Player

For casual entertainment:

Choose lower volatility games

Size bets for session duration

Accept small steady losses

Enjoy the experience

The Advantage Player

For positive EV play:

Volatility determines bankroll requirement

Higher volatility = more capital needed

Must survive to reach long-term

Calculate RoR before committing

Volatility and Strategy

Strategy Adjustments

Volatility doesn't change optimal strategy, but affects:

Practical decisions:

Which games to play given bankroll

Session length expectations

When to quit (not mathematically relevant, but practical)

The Misconception

Some players think:

"I should play more conservatively when losing"

Reality: Optimal strategy is optimal regardless of current session results.

Long-Term Convergence

The Law of Large Numbers

Over time, results approach expected value:

Hands PlayedResult Range (95%)1,000-15% to +14% of expectation10,000-5% to +4%100,000-1.5% to +1%1,000,000-0.5% to +0.5%

The Royal Flush Reality

At 40,000 hands per Royal:

10,000 hands: 22% chance of hitting Royal

40,000 hands: 63% chance of at least one Royal

100,000 hands: 92% chance of at least one Royal

Until Royals normalize, results remain volatile.

Practical Applications

Bankroll Calculator Inputs

Needed for accurate planning:

Game variance (standard deviation)

Expected edge (positive or negative)

Risk tolerance (acceptable RoR)

Session/lifetime goals

Bet sizing flexibility

The Conservative Approach

When uncertain, assume:

Higher variance than calculated

Worse luck than average

Longer to converge

More bankroll needed

Understanding volatility mathematics transforms video poker from a mystery into a manageable mathematical challenge, enabling informed decisions about game selection, bankroll management, and realistic expectations.