Variance and Expected Value in Gambling: Understanding Risk and Reward Mathematics
Every gambling outcome—whether at a casino table, slot machine, or sportsbook—is governed by two fundamental mathematical concepts: expected value and variance. Expected value tells you what happens in the long run, while variance explains why short-term results can look nothing like that prediction. Understanding these concepts transforms gambling from a mystery into mathematics.
According to research published through JSTOR's academic databases, expected value and variance form the foundation of probability theory applied to gambling. This guide explains both concepts in practical terms, demonstrates how to calculate them for any bet, and reveals why they matter for anyone seeking to understand how gambling truly works. For hands-on exploration of these concepts, try our Bankroll Simulator which visualizes variance in action.
The Two Pillars of Gambling Mathematics
Before diving into formulas, consider a simple analogy. Imagine driving on a highway where the speed limit is 60 mph. Expected value is like the average speed of all cars—perhaps 58 mph due to traffic and stops. Variance describes how much individual cars deviate from that average: some travel at 70 mph, others at 45 mph. In gambling, expected value is your mathematical destiny over time, while variance determines the bumps along the way.
Expected Value (EV)
The mathematical average outcome of a bet over infinite repetitions. Tells you exactly how much you'll win or lose per bet in the long run.
Key insight: All casino games have negative EV for players—this is the house edge.
Variance
Measures how spread out results are around the expected value. High variance means wild swings; low variance means consistent results.
Key insight: Variance explains why you can win tonight despite negative EV—and why you'll lose eventually.
Expected Value: Your Mathematical Destiny
Expected value (EV) is the most important number in gambling mathematics. It represents the average outcome of a bet if you could repeat it an infinite number of times. Every casino game, sports bet, and lottery ticket has a calculable expected value that determines whether you profit or lose in the long run.
The Expected Value Formula
Calculating expected value requires knowing all possible outcomes and their probabilities. The formula multiplies each outcome by its probability and sums the results:
Expected Value Formula
EV = Σ(Outcome × Probability)
Or expanded:
EV = (Win Amount × P(Win)) + (Loss Amount × P(Lose))
Example: Roulette Red/Black Bet
Let's calculate the expected value of a $10 bet on red in American roulette. As we explain in our roulette odds analysis, the American wheel has 18 red numbers, 18 black numbers, and 2 green zeros (0 and 00).
EV Calculation: $10 on Red (American Roulette)
Result: You lose $0.53 on average per $10 bet (5.26% house edge)
What Expected Value Tells You
The sign of EV determines whether a bet favors you or the house:
- Positive EV (+EV): Long-term profitable. You make money over time. Found in advantage play situations like card counting or finding mispriced sports bets.
- Negative EV (-EV): Long-term unprofitable. The house profits over time. This describes virtually all standard casino games.
- Zero EV (EV = 0): Break-even over infinite play. Theoretically fair but rarely exists in real gambling due to house margins.
Critical Understanding
Expected value is not what you'll experience in any single session or even a hundred sessions—it's the mathematical center around which your results distribute over time. A negative EV doesn't mean you can't win tonight; it means if you play forever, you'll definitely lose.
EV Across Different Casino Games
Different games have different expected values per dollar bet. The lower the house edge, the closer EV is to zero (break-even). Here's how common games compare, information that aligns with data from the American Gaming Association:
Blackjack (Basic Strategy)
Baccarat (Banker)
Craps (Pass Line)
Roulette (American)
Slots (Average)
Keno
As you can see, blackjack with perfect basic strategy offers the best expected value, while keno has among the worst. This is why game selection matters—playing $1,000 through blackjack costs you about $5, while the same amount through keno costs $250-$400.
Variance: Why Short-Term Results Differ
If expected value tells you where you'll end up, variance tells you about the journey. Variance is a statistical measure of how spread out results are from the average. In gambling, high variance means dramatic swings—big wins and big losses—while low variance means results cluster closer to expected value.
Understanding Variance Visually
Low Variance (Blackjack): Results cluster tightly around EV
High Variance (Slots/Jackpots): Results spread widely around EV
The Variance Formula
Variance is calculated by measuring how far each possible outcome deviates from expected value, squaring those deviations, and taking the probability-weighted average:
Variance Formula
Variance = Σ[P(outcome) × (outcome - EV)²]
Standard Deviation = √Variance
Standard deviation is often more useful because it's in the same units as the original bet.
Example: Comparing Roulette Bets
Consider two different roulette bets, each with the same expected value but vastly different variance:
| Bet Type | Win Probability | Payout | House Edge | Variance |
|---|---|---|---|---|
| Red/Black | 47.37% | 1:1 | 5.26% | Low |
| Single Number | 2.63% | 35:1 | 5.26% | Very High |
Both bets have identical house edge (5.26%), meaning identical long-term expected loss. But the experience differs dramatically:
- Red/Black: You win roughly half your bets. Over 100 spins, your result is relatively predictable—probably down $5-$15.
- Single Number: You lose most bets but occasionally win 35x your stake. Over 100 spins, you might be down $200 or up $500—massive swings despite same EV.
Why Variance Matters for Your Bankroll
Variance directly impacts your risk of ruin—the probability of losing your entire bankroll. According to gambling mathematics research from UNLV's International Gaming Institute, high variance games require larger bankrolls to survive inevitable losing streaks. This is fundamental to understanding how slot machine volatility affects your play.
The Law of Large Numbers
The law of large numbers is why casinos always win despite individual players sometimes walking away with profits. It states that as sample size increases, actual results converge toward expected value. This mathematical principle, documented by the Encyclopedia Britannica, explains everything about why gambling establishments are profitable businesses.
How Many Trials Until Results Converge?
The speed of convergence depends on variance. Low variance games converge faster; high variance games require more trials:
After 100 Blackjack Hands
After 100 Slot Spins
The Casino's Perspective
A casino doesn't care about variance because they see millions of bets daily. With such large sample sizes, their actual results virtually equal expected value. A slot machine with 8% house edge will return almost exactly 92% of money put through it over a year. Individual sessions vary wildly; aggregate results are predictable.
Practical Applications
Choosing Games Based on Your Goals
Understanding EV and variance helps you select games matching your objectives:
- Maximize playing time: Choose low house edge, low variance games (blackjack basic strategy, baccarat banker). Your money lasts longer with predictable results.
- Chase big wins: Choose high variance games (progressive slots, single-number roulette). Accept faster potential loss for lottery-like upside. Learn more about how progressive jackpots work.
- Minimize expected loss: Choose lowest house edge games and avoid high-variance temptations that can deplete bankrolls quickly.
Bankroll Management Implications
Your bankroll requirements scale with variance. The standard recommendation of 20-30 buy-ins for poker or 300-500 base bets for table games exists because of variance. With higher variance games, you need more cushion to survive inevitable downswings while waiting for results to regress toward expected value.
Common Misconception: The Gambler's Fallacy
Understanding variance does NOT mean results "balance out" in a predictable way. After 10 reds in roulette, black is not "due"—each spin remains independent with 47.37% probability. The law of large numbers works through dilution (future results swamping past anomalies), not correction. Past results don't influence future probabilities. We explore this cognitive error in depth in our psychology of gambling article.
Evaluating Betting Systems
EV analysis reveals why betting systems cannot overcome house edge. Systems like Martingale don't change expected value—they only redistribute variance. Doubling after losses creates fewer, larger losing sessions instead of many small ones. The mathematics remains unchanged: negative EV guarantees long-term loss regardless of bet sizing strategy.
Frequently Asked Questions
What is expected value (EV) in gambling?
Expected value is the mathematical average outcome of a bet calculated over infinite repetitions. Find it by multiplying each possible outcome by its probability and summing the results. A coin flip bet paying $2 on heads and losing $1 on tails has EV = (0.5 × $2) + (0.5 × -$1) = $0.50 profit per flip. Positive EV means long-term profit; negative EV means long-term loss. All standard casino games have negative EV due to house edge.
What is variance in gambling?
Variance measures how much actual results deviate from expected value. High variance games like slots produce dramatic swings—big wins alternating with losing streaks. Low variance games like blackjack produce steadier, more predictable results near expected value. Variance explains why short-term outcomes can differ dramatically from mathematical expectation, even though long-term results converge to EV.
Why do short-term gambling results differ from expected value?
Short-term results differ due to variance and the law of large numbers. EV only predicts average outcomes over many trials. In 100 coin flips, you might see 60 heads or 40 heads—both within normal variance. Only over thousands of trials do results converge to expected value. This is why gamblers can win short-term despite negative EV, but mathematics guarantees long-term loss as results regress toward expectation.
How do I calculate expected value for a casino bet?
Multiply each outcome by its probability and sum: EV = Σ(Outcome × Probability). For a $10 roulette bet on red (American wheel): Win = +$10 with probability 18/38; Lose = -$10 with probability 20/38. EV = (18/38 × $10) + (20/38 × -$10) = $4.74 - $5.26 = -$0.53. This negative EV (-5.26% house edge) means you lose $0.53 on average per $10 bet over time.
Can I overcome negative expected value with betting systems?
No. Betting systems like Martingale, Fibonacci, or D'Alembert cannot change expected value—they only redistribute variance. Doubling bets after losses creates fewer but larger losing sessions instead of many small ones. The fundamental mathematics remains: negative EV × total amount wagered = expected loss. No betting pattern can convert a negative EV game into a positive one over time.
Conclusion
Expected value and variance are the twin mathematical forces governing every gambling outcome. Expected value determines your destination—the average result over infinite play—while variance describes the journey, explaining why individual sessions look nothing like that mathematical prediction. Together, they explain why casinos are profitable businesses, why individual gamblers can win in the short term despite unfavorable odds, and why those wins never last indefinitely.
Understanding these concepts won't change the mathematics working against you in casino games. The house edge remains, creating negative expected value on virtually every bet. But this knowledge transforms gambling from mysterious luck into transparent mathematics. You can calculate exactly what any game costs per hour, choose games matching your risk tolerance and goals, and maintain realistic expectations about outcomes.
For those interested in exploring these concepts further, our House Edge Calculator helps compute expected values, while the Bankroll Simulator visualizes variance in action through Monte Carlo simulations. Remember: gambling should be entertainment with a known cost, not a path to profit. Understanding the mathematics helps ensure you stay within that framework.
Disclaimer: This article is for educational purposes only and does not constitute gambling advice. All gambling involves risk of financial loss. Negative expected value means the house edge guarantees long-term player losses. Never gamble with money you cannot afford to lose. If gambling is affecting your life negatively, please visit our responsible gambling resources or contact the National Council on Problem Gambling.