Glossary
Practice Quizzes

7.11 Expected Value

3 min read•june 18, 2024

Gambling games like use to calculate average outcomes over many bets. The formula considers possible outcomes and their probabilities. In roulette, all bets have a negative of -5.26%, meaning players lose money long-term.

Negative expected values favor the casino, creating a . While short-term wins are possible, the means results converge to the expected value over time. Games with lower house edges or skill elements can offer better odds for players.

Expected Value in Gambling

Expected value in roulette bets

Top images from around the web for Expected value in roulette bets

  • Expected Value | Mathematics for the Liberal Arts View original

    Is this image relevant?

  • Probability Reading II | Mathematics for the Liberal Arts View original

    Is this image relevant?

  • Expected Value | Mathematics for the Liberal Arts View original

    Is this image relevant?

1 of 3

Top images from around the web for Expected value in roulette bets

  • Expected Value | Mathematics for the Liberal Arts View original

    Is this image relevant?

  • Probability Reading II | Mathematics for the Liberal Arts View original

    Is this image relevant?

  • Expected Value | Mathematics for the Liberal Arts View original

    Is this image relevant?

1 of 3
  • Expected value formula calculates the average outcome of a bet over many trials: E(X)=∑i=1nxi⋅piE(X) = \sum_{i=1}^{n} x_i \cdot p_i
    • xix_i represents each possible outcome of the bet (win or loss amount)
    • pip_i represents the of each outcome occurring
  • (betting on a specific number like 17) has a of 35 to 1
    • Probability of winning is 138\frac{1}{38} since there is only one winning number out of 38 possible outcomes
    • Probability of losing is 3738\frac{37}{38} as there are 37 non-winning outcomes
    • Expected value for a single number bet: E(X)=(35⋅138)+(−1⋅3738)=−238≈−0.0526E(X) = (35 \cdot \frac{1}{38}) + (-1 \cdot \frac{37}{38}) = -\frac{2}{38} \approx -0.0526 or −5.26%-5.26\%
  • cover multiple numbers and have different payouts and probabilities
    • Red/Black, Even/Odd, or 1-18/19-36 bets have a payout of 1 to 1
      • Probability of winning is 1838\frac{18}{38} as there are 18 winning outcomes out of 38 total
      • Probability of losing is 2038\frac{20}{38} due to 20 non-winning outcomes (including 0 and 00)
      • Expected value for these bets: E(X)=(1⋅1838)+(−1⋅2038)=−238≈−0.0526E(X) = (1 \cdot \frac{18}{38}) + (-1 \cdot \frac{20}{38}) = -\frac{2}{38} \approx -0.0526 or −5.26%-5.26\%
    • Dozens (1-12, 13-24, 25-36) or Columns bets have a payout of 2 to 1
      • Probability of winning is 1238\frac{12}{38} with 12 winning numbers out of 38 total
      • Probability of losing is 2638\frac{26}{38} given 26 non-winning outcomes
      • Expected value for these bets: E(X)=(2⋅1238)+(−1⋅2638)=−238≈−0.0526E(X) = (2 \cdot \frac{12}{38}) + (-1 \cdot \frac{26}{38}) = -\frac{2}{38} \approx -0.0526 or −5.26%-5.26\%

Interpretation of expected value results

  • Expected value represents the average amount won or lost per bet in the long run
    • Negative expected value means the player will lose money on average over many bets
    • Positive expected value indicates the player will win money on average in the long term
  • Casino games like roulette have negative expected values for players, favoring the house
    • House edge is the casino's advantage, represented by the negative expected value
  • Players are likely to lose money in the long run when playing games with negative expected values
    • As more bets are placed, actual results will converge towards the expected value (law of large numbers)
  • measures the variability of outcomes around the expected value

Comparison of gambling scenarios

  • When comparing bets or games, the option with an expected value closest to zero or positive is most advantageous for players
    • In roulette, all bets have the same −5.26%-5.26\% expected value, so no bet is better than others
  • Games with lower house edges (expected values closer to zero) are better for players
    • Blackjack with basic strategy has an expected value around −0.5%-0.5\%, more advantageous than roulette
  • Skill-based games like poker can have positive expected values for skilled players
    • The expected value depends on the player's skill relative to their opponents

Probability and Expected Value in Gambling

  • affects expected value when outcomes are not independent
  • in gambling do not influence each other's probabilities
  • The explains why casino profits tend to be normally distributed over time

Key Terms to Review (27)

Average (or mean) of a set of: The average (or mean) of a set is the sum of all elements in the set divided by the number of elements. It provides a measure of central tendency, representing a typical value within the set.
Binomial distribution: A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. This concept is key in understanding how outcomes can be modeled when there are only two possible results, such as success or failure, which connects to concepts like odds and expected value. It also provides a foundation for approximating distributions that resemble normal distributions under certain conditions.
Blaise Pascal: Blaise Pascal was a French mathematician, physicist, and philosopher, renowned for his contributions to probability theory and combinatorics. His work laid the groundwork for modern statistical analysis and the mathematical understanding of chance, which is crucial for evaluating likelihoods and outcomes in various scenarios. Pascal's Triangle, a simple yet powerful mathematical structure he developed, plays a significant role in combinations and binomial coefficients, influencing how we calculate odds and expected values in probability.
Central Limit Theorem: The Central Limit Theorem states that when independent random variables are added, their normalized sum tends to a normal distribution, even if the original variables themselves are not normally distributed. This powerful concept is foundational in statistics as it allows for the use of the normal distribution to approximate the behavior of sums of random variables, particularly when considering large sample sizes.
Conditional Probability: Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. This concept is essential for understanding how different events can influence one another, especially when using tools like tree diagrams, tables, and outcomes to visualize probabilities, as well as when dealing with permutations and combinations.
Empirical probability: Empirical probability is the probability of an event determined by conducting experiments or observing real-life occurrences. It is calculated as the ratio of the number of favorable outcomes to the total number of trials.
Expected value: Expected value is a fundamental concept in probability that represents the average outcome of a random event over a large number of trials. It is calculated by multiplying each possible outcome by its probability and summing the results.
Expected Value: The expected value, denoted as E(X), is a fundamental concept in probability and statistics that represents the average outcome of a random variable when considering all possible values and their associated probabilities. The formula E(X) = Σ(x * P(x)) calculates this average by summing the products of each outcome (x) and its probability (P(x)). This concept helps in making informed decisions based on the anticipated average results of uncertain events.
Gambling strategies: Gambling strategies refer to various techniques and methods that players use to maximize their chances of winning or to minimize their losses while gambling. These strategies can involve mathematical calculations, psychological tactics, or game theory principles, often aiming to exploit the probabilities of different games. Understanding these strategies is crucial for evaluating the expected value of bets and making informed decisions in gambling scenarios.
Group bets: Group bets refer to a betting strategy where multiple individuals pool their resources together to place a larger bet than they could individually. This approach allows participants to share both the risks and rewards associated with the bet, potentially increasing the overall expected value of their wagers. Group betting is often utilized in various contexts, such as sports betting or lottery pools, where collective participation can enhance the chances of winning.
House edge: House edge is the mathematical advantage that a gambling establishment, like a casino, has over its players. This edge ensures that, over time, the casino will make a profit from the games they offer, regardless of individual player outcomes. The house edge is expressed as a percentage and varies between different games, influencing the expected value of bets made by players.
Independent events: Independent events are outcomes in probability that do not influence each other; the occurrence of one event does not change the probability of the other event occurring. Understanding independent events is crucial for calculating probabilities accurately, especially when using methods like permutations and combinations, assessing odds, applying addition and multiplication rules, evaluating conditional probabilities, and computing expected values.
Insurance premiums: Insurance premiums are the amounts paid periodically to an insurance company by an individual or business in exchange for coverage against potential financial losses. These payments are crucial for maintaining the policy and ensuring that the insured can receive benefits in the event of a claim. Understanding how premiums work is essential for effective budgeting and evaluating expected costs related to potential risks.
Law of large numbers: The law of large numbers is a statistical theorem that states that as the number of trials or observations increases, the sample mean will get closer to the expected value or population mean. This principle highlights the reliability of averages and ensures that larger samples provide a more accurate representation of the overall population, making it essential in probability and statistics.
Mean: The mean, commonly known as the average, is a measure of central tendency that summarizes a set of values by calculating the sum of the values divided by the number of values. It serves as a critical statistical tool that helps in understanding data distributions and making predictions, especially in contexts involving probability and real-world applications like economics or social sciences.
Normal Distribution: Normal distribution is a statistical concept that describes how data points are spread out around the mean, forming a symmetric, bell-shaped curve. This curve illustrates that most observations cluster around the central peak, with probabilities tapering off symmetrically on either side, making it essential for understanding probability and variability in data analysis.
Normal distributions: A normal distribution is a probability distribution that is symmetric around the mean, showing that data near the mean are more frequent in occurrence. It forms a bell-shaped curve where most of the observations cluster around the central peak.
Payout: A payout refers to the amount of money that is distributed to individuals or entities as a result of a financial event, such as winning a bet, receiving an insurance claim, or being paid from an investment. Understanding payouts is crucial when assessing risk and calculating potential returns, especially in scenarios that involve uncertainty and probability.
Pierre de Fermat: Pierre de Fermat was a French lawyer and mathematician who made significant contributions to number theory and probability. He is best known for Fermat's Last Theorem and his work on the foundations of modern probability theory, particularly in the realm of expected value and combinatorics, influencing how we understand odds and chances in mathematical contexts.
Probability: Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. It connects various mathematical concepts by providing a framework to assess and quantify uncertainty in different scenarios, helping to determine outcomes based on different arrangements, selections, and occurrences.
Random variable: A random variable is a numerical outcome of a random process, serving as a way to quantify uncertainty. It can take on different values based on the outcomes of a particular experiment or event, and is often used to model real-world scenarios. Random variables can be classified into two types: discrete and continuous, and they are foundational to understanding probability distributions and statistical measures.
Roulette: Roulette is a popular casino game that involves a spinning wheel with numbered slots and a small ball. Players place bets on where they believe the ball will land when the wheel stops spinning. The game is often used to illustrate concepts of probability and expected value in gambling contexts, helping players understand the odds associated with different types of bets.
Sample space: Sample space is the set of all possible outcomes in a probability experiment. It provides a comprehensive list of everything that could happen during the experiment.
Sample Space: A sample space is the set of all possible outcomes of a random experiment. Understanding the sample space is crucial because it forms the foundation for calculating probabilities, counting outcomes, and analyzing events in various contexts.
Single Number Bet: A single number bet is a specific type of wager in gambling, particularly in games like roulette, where a player bets on one particular number to win. This type of bet has a higher payout than many other bets because of the lower probability of winning, making it a high-risk, high-reward option. Players often use this strategy to maximize their potential winnings, but it's important to understand the odds involved.
Standard Deviation: Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. It indicates how much individual data points deviate from the mean, helping to understand the distribution and spread of data. A low standard deviation means that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. This concept is crucial for interpreting expected values, analyzing central tendencies like the mean, median, and mode, and assessing data distributions, including normal distributions.
Variance: Variance is a statistical measure that represents the degree of spread or dispersion of a set of values around their mean. It helps quantify how much the values in a data set deviate from the average, providing insight into the consistency and variability of the data. Understanding variance is essential in probability, distributions, and regression analysis as it influences predictions and expectations derived from data.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Glossary
Glossary
Next