Introduction

Expected value is a concept used across various fields, from finance and investing to sports betting and decision-making. Simply put, expected value is the average outcome of a certain event, weighted by the probability of each possible outcome occurring. As a tool to quantify uncertainty, expected value helps decision-makers estimate the value or cost of potential actions or decisions by weighing the potential costs and benefits against their likelihood of happening.

Whether calculating expected returns in investing or expected values in sports betting, the concept of expected value remains the same. In this article, we will explore the basics of expected value, including probability, probability models, and decision trees. We will also delve into how expected value plays a pivotal role in investing, sports betting, and decision-making, and provide examples of how it can be applied across different fields.

Understanding Probability

Probability is the likelihood of an event happening, expressed as a fraction or percentage. For instance, if you flip a coin, the probability of getting heads or tails is 50-50 or 0.5. Probability governs the occurrence of events in a random or uncertain setting, which may also be affected by external factors such as luck, skill, or information.

Probability models, such as the binomial and normal distributions, are used to describe the likelihood of events happening under certain conditions or assumptions. For example, if you have a fair coin and flip it ten times, the binomial distribution can be used to calculate the probability of getting a certain number of heads or tails. Similarly, the normal distribution explains the probability of a range of outcomes occurring within a certain confidence interval.

Expected value can be calculated using probability models, often with the help of mathematical or statistical tools such as Excel or Python. The formula for expected value is as follows:

Expected value = probability of outcome 1 x value of outcome 1 + probability of outcome 2 x value of outcome 2 + …

For instance, let’s assume we play a game where we roll a die and receive different amounts of money depending on the outcome. If we pay $1 to play and can either receive $2, $3, $4, $5, or $6, each with equal probability of 1/6, the expected value of this game would be:

Expected value = (1/6 x 2) + (1/6 x 3) + (1/6 x 4) + (1/6 x 5) + (1/6 x 6) = $3.50

This means that on average, we would expect to earn $3.50 per game in the long run if we play multiple times. Expected value can also be negative, indicating a loss, or zero, indicating a break-even situation.

Expected Value in Investing

Expected value is a critical concept in investing, where it is used to estimate potential returns and risks of different investment strategies or assets. By calculating expected returns based on probability models, investors can make informed decisions based on the expected value of each investment option, factoring in the potential for positive or negative outcomes.

Using expected value for risk management is also crucial in investing, as it helps investors set realistic expectations for returns and avoid overreliance on short-term gains or losses. Portfolio diversification, which aims to reduce risks by investing in different asset classes or sectors, can also be guided by expected value calculations, by comparing the expected return and standard deviation of each asset to optimize the portfolio’s return and risk profile.

For example, suppose you are considering investing in two mutual funds, A and B, each with a potential return and standard deviation as shown below:

Assuming that these are the only two options you have for investing, you can calculate the expected value of each fund as:

Expected value of Fund A = 0.5 x (10% – 15%) + 0.5 x (10% + 15%) = 10%

Expected value of Fund B = 0.5 x (8% – 10%) + 0.5 x (8% + 10%) = 8%

Although Fund A has a higher expected return, it also has a higher standard deviation, indicating greater volatility and potential risks. Based on your risk tolerance and investment goals, you can use expected value calculations to choose the best investment strategy that balances risk and return.

Sports Betting and Expected Value

Expected value can also be applied to sports betting, where it is used to evaluate the potential value of different bets and make informed decisions based on the expected outcome and payout. By calculating the expected value of each bet, bettors can identify opportunities where the expected return is higher than the initial cost of the bet, known as positive expected value (EV+).

Understanding expected value in sports betting requires a good grasp of probability and statistics, as well as the different types of bets and their corresponding odds. For instance, in a coin toss, the odds of heads or tails are 1/2 or 50-50. However, if a bettor bets $10 on heads and wins, they receive only $9 as payout, not the full $10. This discrepancy in payout and actual odds can create opportunities for positive expected value bets, where the payout is higher than what is expected based on the odds.

To see how this works, suppose there is a soccer match between Team A and Team B, and a bettor bets $100 on Team A to win, with odds of 1.8. If Team A wins the match, the bettor receives $180 ($100 initial bet + $80 payout). However, if Team A loses, the bettor loses $100. Based on the probability of Team A winning, the expected value of this bet can be calculated as:

Expected value of Team A bet = 0.6 x (180) + 0.4 x (-100) = $68

This means that the expected value of the bet is positive, indicating that it is a good bet to make from a probability and payout perspective.

Strategies for making smarter bets with expected value include analyzing the odds offered by different bookmakers, avoiding emotional or impulsive betting decisions, and keeping track of your results to evaluate your betting strategy’s effectiveness.

Expected Value in Decision-Making

Expected value is not limited to investing or sports betting but can also be applied to decision-making in different fields. By weighing the potential costs and benefits of different options, decision-makers can estimate the expected value of each option, taking into account uncertainty, risk, and other variables.

One example of applying expected value in decision-making is decision trees. Decision trees are used to model the different paths and outcomes of a decision by breaking it down into smaller, more manageable steps. By assigning probabilities to each step and its outcomes, decision-makers can use expected value to evaluate the expected value and expected utility of each decision option.

Utility theory, which measures the value or desirability of different outcomes, can also be used in conjunction with expected value to make better decisions. By comparing the expected utility of different options, decision-makers can factor in subjective preferences and biases, such as risk aversion or loss aversion.

For instance, suppose you are a company manager deciding whether to invest in a new product line or stay with the current one. By creating a decision tree for each option and assigning probabilities and values to the different outcomes, you can use expected value to compare the expected payoff and expected value of each option. Utility theory can also be used to factor in other considerations, such as the opportunity cost of not investing in a potentially more profitable line, or the potential losses if the new line fails to gain traction.

Case Study Examples

Expected value can be applied to various fields, including finance, business, psychology, and others. Here are some examples of how expected value is used in each field:

• In finance, expected value is used to calculate expected returns and risks of different investment strategies or assets and optimize portfolio diversification.
• In business, expected value is used in decision-making to evaluate the potential benefits and costs of different options and inform resource allocation and risk management strategies.
• In psychology, expected value is used in utility theory to understand how people weigh the costs and benefits of different decisions and behavior, and how decision-making can be influenced by risk perception and biases.
• In health, expected value is used to inform medical decision-making and cost-effectiveness analysis, where the expected value of different treatments and interventions is compared based on their potential health benefits and costs.

Conclusion

Expected value is a powerful tool for estimating the potential costs and benefits of different decisions or actions, under uncertainty and risks. From investing and sports betting to decision-making across different fields, the concept of expected value can be applied to calculate expected returns, identify opportunities, and balance risks. By understanding probability, probability models, and decision trees, decision-makers can make better decisions by using expected value to weigh the potential outcomes and take into account different variables and considerations.