Expected value (EV) is a fundamental concept in probability theory and statistics that plays a crucial role in gambling.
At its core, EV represents the average outcome of an event if it were to be repeated many times.
In gambling, understanding and applying the concept of expected value can be the difference between making informed decisions and falling prey to common pitfalls. For gamblers, whether casual or professional, grasping the concept of expected value is essential. It provides a mathematical framework to evaluate the potential profitability of bets and games over the long term.
By calculating the EV of a particular wager or game, players can make more rational choices about where to place their bets and how to manage their bankroll. This article on Casinoz will explore the intricacies of expected value in gambling, its calculation, application across various casino games, and its implications for decision-making. We'll examine how casinos use the EV to maintain their edges and how advantage gamblers can use this concept to identify potentially profitable opportunities.
The Concept of Expected Value
Expected value (EV) is a statistical concept that represents the average outcome of an experiment if it were to be repeated infinitely.
In mathematical terms, it's the sum of each possible outcome multiplied by its probability of occurrence.
The expected value is expressed as EV = Σ (Outcome * Probability), where Σ represents the sum of all possible outcomes.
To illustrate this concept, let's consider a simple example:
Imagine a fair coin toss where you win $1 for heads and lose $1 for tails. The expected Value of this game would be:
EV = (1 * 0.5) + (-1 * 0.5) = 0.5 - 0.5 = 0
This means that, on average, you would expect to break even if you played this game many times.
Now, let's consider a slightly more complex example:
Suppose you're offered a game where you roll a six-sided die. If you roll a 6, you win $10. For any other number, you lose $2. What's the expected value of this game?
EV = (10 * 1/6) + (-2 * 5/6) = 1.67 - 1.67 = 0
Again, this game has an expected value of 0, meaning it will be fair in the long run.
Understanding these basic examples helps lay the foundation for applying Expected Value to more complex gambling scenarios. In the following sections, we'll explore how to calculate and interpret expected value in various casino games and betting situations.
Calculating Expected Value in Gambling
Calculating the expected value is crucial for making informed decisions in gambling. The general formula remains the same as in our basic examples, but it's often applied to more complex scenarios.
General formula: EV = (Probability of Winning * Amount Won per Bet) + (Probability of Losing * Amount Lost per Bet)
Let's walk through a step-by-step process with a realistic gambling example:
- Step 1: Identify the possible outcomes.
- Step 2: Determine the probability of each outcome.
- Step 3: Calculate the potential winnings or losses for each outcome.
- Step 4: Multiply each outcome by its probability.
- Step 5: Sum all the results.
Example: Roulette bet on red. Let's calculate the EV of betting $10 on red in American roulette:
- Step 1: Possible outcomes: Win (red) or Lose (black or green).
- Step 2: Probabilities: The probability of red is 18/38 (there are 18 red numbers out of 38 total). The probability of not red (losing): 20/38 (18 black + 2 green).
- Step 3: Potential winnings/losses: If you win, you get your $10 back plus $10 profit. If you lose, you lose your $10 bet.
- Step 4: Multiply outcomes by probabilities: Winning: (18/38) * $10 = $4.74. Losing: (20/38) * (-$10) = -$5.26.
- Step 5: Sum the results: EV = $4.74 + (-$5.26) = -$0.52.
The negative expected value (-$0.52) indicates that, on average, you can expect to lose 52 cents for every $10 bet on red in American roulette over the long run.
This calculation method can be applied to various gambling scenarios, helping players understand the long-term implications of their bets.
Expected Value in Casino Games
Different gambling games have varying expected values due to their unique rules and probabilities. Let's examine some popular casino games and their EV characteristics:
Basic strategy play:
- EV varies from -0.5% to +0.5%, depending on specific rules.
- Card counting can potentially shift EV in the player's favor.
Example: In a game with favorable rules and perfect basic strategy, EV ≈ -$0.005 per $1 bet.
| Name | Soft | Return to player | ||
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99.78% | ||
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99.69% | ||
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99.65% | ||
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99.6% | ||
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99.59% | ||
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99.59% | ||
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99.54% | ||
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99.54% | ||
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99.54% | ||
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99.33% |
American Roulette (38 numbers including 0 and 00):
- Single number bet: EV = (-$0.053 per $1 bet).
- Red/Black bet: EV = (-$0.053 per $1 bet).
European Roulette (37 numbers including only 0):
- Single number bet: EV = (-$0.027 per $1 bet).
- Red/Black bet: EV = (-$0.027 per $1 bet).
Note: European Roulette has a better EV because it has only one zero.
EV is highly variable depending on the skill level of players.
In cash games, skilled players can achieve positive EVs.
Tournament poker introduces additional variables affecting EV.
- Pass/Come bets: EV ≈ -$0.014 per $1 bet.
- Don't Pass/Don't Come: EV ≈ -$0.013 per $1 bet.
Proposition bets generally have much worse EVs.
- Banker bet: EV ≈ -$0.0106 per $1 bet.
- Player bet: EV ≈ -$0.0124 per $1 bet.
- Tie bet: EV ≈ -$0.1436 per $1 bet.
Generally, they have negative EVs for players.
- Modern slots: EV typically ranges from -2% to -15%.
Example: $1 slot with a 95% payback rate has EV = -$0.05 per $1 bet.
| Name | Return to player | Max payout | Variance | Soft | ||
| 99.07% | — | Low |
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| 98.72% | — | — |
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| 98.67% | — | — |
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| 98.6% | — | — |
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| 98.18% | — | Low |
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| 98.13% | x20 € | — |
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| 98.12% | x500 $ | Low |
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| 98.11% | x1500 $ | High |
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| 98.11% | x7500 $ | — |
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| 98.1% | x2500 $ | — |
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| 98.1% | x2500 $ | — |
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| 98.06% | x5000 $ | High |
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| 98.03% | x15000 $ | Middle |
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| 98.01% | x500 $ | — |
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| 98% | x2000 € | Middle |
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Understanding the EVs of different games and bets within those games can help gamblers make more informed decisions about where to place their money. It's important to note that these EVs represent long-term averages, and short-term results can vary significantly.
Positive vs. Negative Expected Value
Understanding the difference between positive and negative Expected Value (EV) is crucial for gamblers seeking to make informed decisions.
Positive Expected Value
A bet or game with a positive EV is expected to be profitable in the long run.
Example: If a bet has an EV of +$0.10 per $1 wagered, you would expect to profit 10 cents for every dollar bet over many repetitions.
Positive EV opportunities are rare in casino games but can sometimes be found in skill-based games like poker or sports betting.
Negative Expected Value
Most casino games and bets have a negative EV, meaning the player is expected to lose money over time.
Example: The American Roulette bet on red has an EV of -$0.053 per $1 bet, meaning you're expected to lose 5.3 cents per dollar wagered in the long run.
How casinos maintain an edge:
- House Edge: This is the built-in advantage that ensures the casino's profit over time. It's essentially the negative EV of the games from the player's perspective. Example: In American Roulette, the house edge is 5.26% on most bets.
- Game Design: Casinos carefully design games to have a negative EV for players. They balance payouts and probabilities to ensure a profitable outcome for the house.
- Rules and Restrictions: Casinos implement rules that limit players' ability to gain an advantage. Example: Limiting deck penetration in blackjack to make card counting less effective.
- Volume and Time: Even with a small edge, casinos profit by encouraging high-volume play over extended periods.
- Psychological Factors: Casinos use various techniques to encourage continued play, such as complimentary services and creating an exciting atmosphere.
Understanding positive and negative EVs helps gamblers recognize that most casino games are designed to favor the house. While short-term wins are possible, the law of large numbers ensures that the casino's edge will prevail over time.
Using Expected Value to Make Decisions
Understanding Expected Value (EV) can significantly improve a gambler's decision-making process. Here's how to apply EV concepts when gambling:
- Evaluating Bets: Calculate the EV of different bets within a game. Choose bets with the highest (or least negative) EV when possible. Example: In Baccarat, the Banker bet (EV ≈ -1.06%) is slightly better than the Player bet (EV ≈ -1.24%).
- Game Selection: Compare the EV of different games. Play games with lower house edges (less negative EV). Example: Choose European Roulette (house edge 2.7%) over American Roulette (house edge 5.26%).
- Skill-Based Games: Your skill can influence the EV in games like poker or blackjack. Invest time in learning optimal strategies to improve your EV. Example: Using the perfect basic strategy in blackjack can reduce the house edge to less than 0.5% under favorable rules.
- Bonus and Promotion Evaluation: Calculate the EV of casino bonuses and promotions. Some offers may temporarily shift the EV in the player's favor. For example, a 100% match bonus on a deposit might create a positive EV situation if the wagering requirements are reasonable.
- Bankroll Management: Use EV to determine appropriate bet sizes relative to your bankroll. The Kelly Criterion, which uses EV in its formula, can guide optimal bet sizing for positive EV situations.
- Long-term vs. Short-term Outcomes: Understand that EV represents long-term expectations. Due to variance, short-term results can deviate significantly from EV. Make decisions based on long-term EVs rather than recent results.
- Quitting Decisions: Use EV to determine when to stop playing. If all available options have negative EVs, consider ending the gambling session.
- Comparing Gambling to Other Activities: Use EV to compare gambling to other forms of entertainment or investment.
Consider the entertainment value alongside the mathematical expectation. By consistently making decisions based on EV, gamblers can minimize losses in negative EV games and maximize profits in rare positive EV situations. However, it's crucial to remember that even with optimal decision-making, most casino games will have a negative EV for the player in the long run.
Common Misconceptions about Expected Value
Despite its importance, gamblers often misunderstand expected value. Here are some common misconceptions.
The Gambler's Fallacy
- Misconception: Past outcomes affect future probabilities in independent events.
- Reality: In games of chance, each event is independent. Previous outcomes do not influence future results.
Example: Believing that after several reds in roulette, black is "due" to come up.
Misunderstanding Short-term vs. Long-term Results
- Misconception: EV predicts short-term outcomes.
- Reality: EV is a long-term average. Short-term results can vary significantly.
Example: A slot machine with a 95% payback rate won't necessarily return $95 for every $100 wagered in a single session.
Ignoring Variance
- Misconception: Games with the same EV are equally risky.
- Reality: Variance (the spread of possible outcomes) can differ greatly between games with the same EV.
Example: A low-volatility slot machine and a high-stakes lottery might have the same EV but very different risk profiles.
Overestimating Skill Impact
- Misconception: Skill can overcome negative EV in all games.
- Reality: While skill can improve EV in some games, it cannot overcome the house edge in purely chance-based games.
Example: No amount of skill can make roulette a positive EV game for the player.
Misapplying EV to Single Events
- Misconception: EV guarantees a specific outcome for a single bet.
- Reality: EV is an average over many repetitions, not a prediction for a single event.
Example: A positive EV bet can still lose, and a negative EV bet can still win in the short term.
Neglecting the Time Factor
- Misconception: A small negative EV is insignificant.
- Reality: Even small negative EVs compound over time and with increased play.
Example: A -0.5% EV might seem small but can lead to significant losses over thousands of bets.
Misunderstanding Probability
- Misconception: All outcomes in a game are equally likely.
- Reality: Probabilities can vary greatly between different outcomes.
Example: In craps, a roll of 7 (probability 1/6) is much more likely than a roll of 2 (probability 1/36).
Believing in "Due" Payouts
- Misconception: Machines or games that haven't paid out recently are "due" for a win.
- Reality: Each play is independent, and games don't "remember" past results.
Example: A slot machine that hasn't hit a jackpot in a long time is not more likely to pay out soon.
Apprehending these misconceptions can help gamblers avoid common pitfalls and make more rational decisions based on true EV principles.
Advanced Math Concepts in Gambling
While understanding basic expected value is crucial, there are more advanced concepts that can provide deeper insights into gambling mathematics:
Variance and Standard Deviation
Variance measures the spread of possible outcomes around the expected value. Standard deviation is the square root of variance and is often used to quantify risk. Higher variance games can have more extreme short-term results, even if the EV is the same.
Example: Comparing low vs high volatility slots with the same RTP (Return to Player).
Risk of Ruin
This concept calculates the probability of losing your entire bankroll. Factors include your starting bankroll, bet size, and the game's odds. Understanding risk of ruin can help with better bankroll management.
Formula: R = ((q/p)^N - 1) / ((q/p)^A - 1), where:
- p = probability of winning a bet;
- q = 1 - p (probability of losing);
- N = number of bets your bankroll can sustain;
- A = number of bets needed to reach your goal.
Kelly Criterion
A formula used to determine the optimal bet size for positive EV situations.
Kelly fraction = (bp - q) / b, where:
- b = the net odds received on the bet;
- p = probability of winning;
- q = probability of losing (1 - p).
It is often used in sports betting and poker.
Compound Probability
It is understanding how probabilities combine in multi-step or continuing bet scenarios.
Example: Calculating the probability of winning multiple hands in a row in blackjack.
Expected Value of Variable Outcomes
It is calculating EV when there are multiple possible outcomes, each with different probabilities and payouts.
It is common in poker and some proposition bets.
Game Theory and Mixed Strategies
It is applicable in competitive games like poker.
It involves calculating optimal frequencies for different actions to maximize EV against intelligent opponents.
Regression to the Mean
It is the statistical tendency for extreme results to be followed by more average results.
It is important for understanding why "hot" or "cold" streaks don't predict future outcomes.
Conditional Probability
It is calculating probabilities when some information is known.
Example: Adjusting blackjack strategy based on exposed cards.
These advanced concepts provide a more comprehensive understanding of gambling maths. They're particularly useful for advantage gamblers, game designers, and those interested in the deeper mathematical aspects of gambling.
Practical Applications of EV
Understanding the expected value and its related concepts has several practical applications in gambling.
Bankroll Management
Use EV to determine appropriate bet sizes relative to your bankroll.
Example: Limit bets to a small percentage of your bankroll on negative EV games to extend playing time.
Apply the Kelly Criterion or a fractional Kelly approach for positive EV situations.
Game Selection
Choose games and bets with the best (least negative) EV.
Example: Opt for European Roulette over American Roulette, or Pass Line bets over proposition bets in Craps.
In skill-based games like poker, select tables where you have a skill advantage, potentially creating positive EV situations.
Bonus Hunting
Evaluate casino bonuses and promotions using EV calculations. Determine if a bonus offer provides positive EV considering wagering requirements and game restrictions.
Example: A 100% match bonus with 20x wagering requirement on slots with 97% RTP could be +EV.
Sports Betting
Use EV to identify value bets where the bookmaker's odds don't accurately reflect true probabilities.
Apply the Kelly Criterion to optimize bet sizing in positive EV situations.
Poker Strategy
Use EV calculations to make optimal decisions in different situations.
Example: Deciding whether to call, fold, or raise based on pot odds and implied odds.
Advantage Gambling Techniques
In games like blackjack, use EV to determine when conditions are favorable for increasing bets (e.g., card counting).
Evaluate the EV of shuffle tracking, hole carding, or other advantage techniques.
Gambling Budget Planning
Use EV to estimate expected losses for a gambling session or vacation.
Example: If playing slots with a 5% house edge, budget for an expected loss of $50 per $1000 wagered.
Comparing Gambling to Other Activities
Use EV to compare the cost of gambling to other forms of entertainment.
Consider the entertainment value alongside potential financial outcomes.
Developing Betting Systems
Use EV to evaluate the long-term viability of betting systems.
Understand that no betting system can overcome a negative EV in the long run.
Risk Assessment
Use variance and standard deviation alongside EV to understand the risk profile of different games or bets.
Make informed decisions about acceptable levels of risk.
By applying these practical uses of EV, gamblers can make more informed decisions, manage their bankrolls more effectively, and potentially identify advantageous situations.
Limitations of Expected Value in Gambling
While Expected Value (EV) is a powerful tool for analyzing gambling situations, it's important to understand its limitations.
- Short-term Variance: EV represents long-term averages, but short-term results can deviate significantly. Example: A negative EV game can still produce wins in the short term, and vice versa.
- Emotional Factors: EV doesn't account for the psychological aspects of gambling, such as excitement, fear, or tilt. These emotional factors can lead to irrational decisions that deviate from optimal EV-based play.
- Incomplete Information: In some games, like poker, you don't always have complete information about probabilities. This can make precise EV calculations challenging or impossible in real-time situations.
- Skill Variability: In skill-based games, EV calculations assume consistent performance. Factors like fatigue, distraction, or varying skill levels can affect actual outcomes.
- Dynamic Environments: In games with changing conditions (e.g., progressive jackpots), EV can shift over time. Constant recalculation may be necessary but not always practical.
- Oversimplification: Basic EV models may not capture all nuances of complex games or betting scenarios. Simplified EV calculations might miss important factors that influence outcomes.
- Bankroll Constraints: EV doesn't account for the risk of ruin or the impact of limited bankrolls. Optimal play based on EV might not be feasible if it risks depleting your entire bankroll.
- Time Horizons: The "long run" implied in EV calculations may be longer than a player's lifetime of play. Theoretical EV might never be realized in practice due to limited play.
- Non-monetary Factors: EV typically focuses on financial outcomes and doesn't account for entertainment value or other non-monetary benefits.
- Assumption of Rationality: EV-based strategies assume players always make rational decisions. In reality, cognitive biases and emotions often influence gambling behavior.
- House Edge Evolution: Casinos may adjust rules or payouts over time, changing the EV of games. Historical EV calculations may become outdated.
- Individual Variation: EV represents an average across all players, but individual results can vary widely. Some players may consistently perform above or below EV due to skill or luck.
While EV remains a valuable tool for analysis and decision-making, it should be used with other considerations, including risk tolerance, bankroll management, and personal enjoyment.
| Casino | Bonuses | Editors rating | |||
| 100% to 1000 $ x35 |
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| 125% to 80 $ x35 |
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| 100% to 400 $ x50 |
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Conclusion
Expected value (EV) is a fundamental concept in gambling mathematics that provides a powerful framework for analyzing games of chance and making informed decisions.
Understanding EV is essential for responsible gamblers. It demystifies the mathematics behind gambling, helping players make more rational decisions and set realistic expectations.
However, it's crucial to remember that even with perfect EV-based play, most casino games will still favor the house in the long run.
Ultimately, while EV is an invaluable tool for analysis and strategy, gambling should primarily be viewed as entertainment. Players should always gamble responsibly, setting strict limits on time and money spent and never risking more than they can afford to lose.
