Mastering Expected Value: A Step-by-Step Guide with Real-World Examples

Expected value (EV) is a fundamental concept in probability and decision-making. It’s the average outcome you can anticipate from a probabilistic event, considering both the potential gains and losses, along with their respective likelihoods. Whether you’re a seasoned investor, a casual gambler, or simply someone making everyday choices, understanding how to calculate expected value can significantly enhance your decision-making process. This article will provide a comprehensive, step-by-step guide on how to calculate expected value, along with clear explanations and practical examples to solidify your understanding.

What is Expected Value?

At its core, expected value represents the long-term average result of repeating an experiment or event multiple times. It doesn’t tell you what will happen in any single instance, but rather provides a weighted average of all possible outcomes. Imagine flipping a fair coin multiple times. You know that each flip has a 50% chance of landing on heads and 50% chance of landing on tails. While you might not get exactly 50 heads and 50 tails in 100 flips, if you flipped the coin an infinite number of times, the percentage would approach 50%. Expected value extends this concept to scenarios with more complex outcomes and associated probabilities.

Think of expected value as a tool to make informed choices under uncertainty. By quantifying potential risks and rewards, you can evaluate opportunities objectively and avoid decisions driven by gut feelings alone. It helps you assess the fairness of a game, compare investment options, and even make better personal choices.

The Formula for Expected Value

The formula for calculating expected value is surprisingly simple:

EV = (Outcome 1 * Probability 1) + (Outcome 2 * Probability 2) + … + (Outcome n * Probability n)

Where:

• EV is the Expected Value
• Outcome represents each possible result of an event (this can be a gain or a loss, represented by a positive or negative number, respectively)
• Probability represents the likelihood of each outcome occurring (expressed as a decimal or a fraction between 0 and 1)
• n represents the total number of distinct outcomes

Essentially, you multiply each possible outcome by its corresponding probability, and then you sum up all those results. The final sum is your expected value.

Step-by-Step Guide to Calculating Expected Value

Let’s break down the calculation process into easy-to-follow steps:

1. Identify All Possible Outcomes: The first and most crucial step is to identify all the possible results of the event you are analyzing. Be thorough and make sure you do not miss any potential outcome. This may involve listing possible wins, losses, or neutral events.
2. Determine the Value of Each Outcome: Assign a numerical value to each identified outcome. This is usually a monetary value, but it could also be anything that represents the ‘value’ of the outcome in the specific context of the problem (e.g., points, time saved, quality improvement). Make sure to use consistent units for all values (e.g. all in dollars or all in points). Remember that losses should be represented by negative numbers, and gains by positive numbers.
3. Determine the Probability of Each Outcome: For each outcome, calculate or determine its probability of occurrence. Probabilities are always values between 0 and 1, inclusive, or sometimes are expressed as percentages (0-100%). Probability 0 means there’s no chance of the outcome happening and probability 1 means the outcome is guaranteed to happen. The sum of the probabilities for all possible outcomes in a single event must always equal 1 (or 100%).
4. Multiply Outcome Values by their Probabilities: Multiply the numerical value of each outcome by its corresponding probability. For instance, if one outcome is winning $100 with a probability of 0.2, the calculation is $100 * 0.2 = $20.
5. Sum the Results: Add up all the products you calculated in the previous step. The result of this summation is the expected value.

Examples of Expected Value Calculations

To solidify your understanding, let’s explore some examples:

Example 1: A Simple Coin Toss Game

Let’s say you’re playing a game where you flip a fair coin. If it lands on heads, you win $5. If it lands on tails, you lose $3. What is the expected value of playing this game?

1. Identify all possible outcomes: Heads (win $5) or Tails (lose $3).
2. Determine the value of each outcome: Win is +$5; Loss is -$3.
3. Determine the probability of each outcome: The probability of heads is 0.5, and the probability of tails is 0.5 (assuming a fair coin).
4. Multiply outcome values by their probabilities: For Heads: $5 * 0.5 = $2.5. For Tails: -$3 * 0.5 = -$1.5
5. Sum the results: $2.5 + (-$1.5) = $1

The expected value of playing this game is $1. This means that, on average, you can expect to win $1 for each game you play in the long run. It’s important to understand that you will never win exactly $1 in any single game – you will either win $5 or lose $3, but over many plays, your average winnings will converge towards $1.

Example 2: A Raffle Ticket

Imagine you buy a raffle ticket for $2. There are 100 tickets in total. The prize is $100. What is the expected value of buying one ticket?

1. Identify all possible outcomes: Winning the $100 prize or Losing the $2 cost of the ticket.
2. Determine the value of each outcome: Winning: $100 – $2 (cost of ticket) = $98; Losing: -$2.
3. Determine the probability of each outcome: Winning Probability: 1/100 = 0.01; Losing probability: 99/100 = 0.99
4. Multiply outcome values by their probabilities: Winning: $98 * 0.01 = $0.98; Losing: -$2 * 0.99 = -$1.98.
5. Sum the results: $0.98 + (-$1.98) = -$1

The expected value of buying one raffle ticket is -$1. This means that, on average, for every ticket you buy, you can expect to lose $1 in the long run. This helps explain why these games are often profitable for the organizers, but not usually for the players. This does not, however, mean it’s guaranteed you will lose a dollar each time, but in the long run, that is the expected loss per play.

Example 3: A More Complex Business Decision

A company is considering launching a new product. Market research suggests three possible scenarios:

• High Success: 40% chance of making $500,000 in profit
• Moderate Success: 30% chance of making $200,000 in profit
• Failure: 30% chance of losing $100,000.

Should the company proceed with the product launch? Let’s calculate the expected value:

1. Identify all possible outcomes: High success, Moderate success, Failure
2. Determine the value of each outcome: High success: +$500,000; Moderate success: +$200,000; Failure: -$100,000.
3. Determine the probability of each outcome: High success: 0.40; Moderate success: 0.30; Failure: 0.30.
4. Multiply outcome values by their probabilities: High success: $500,000 * 0.4 = $200,000; Moderate success: $200,000 * 0.3 = $60,000; Failure: -$100,000 * 0.3 = -$30,000.
5. Sum the results: $200,000 + $60,000 + (-$30,000) = $230,000.

The expected value of launching the product is $230,000. This is a positive number indicating that the company can expect an average profit of $230,000 if they launch the product, considering all the possible outcomes and their probabilities. This analysis would support moving forward with the launch. It’s important to note that they could still lose money if the failure scenario unfolds, but based on all the available data, the probability of this is outweighed by the potential gains.

Example 4: Insurance

An insurance company offers a policy that pays out $500,000 in the event of an accident, but the policy costs $1000 per year. The probability of the accident happening each year is 0.1%. What is the expected value for the insurance company?

1. Identify all possible outcomes: Paying out $500,000 (accident occurs) or collecting the $1000 payment and not paying out (no accident occurs).
2. Determine the value of each outcome: Paying out the claim: -$500,000 + $1000 (premium) = -$499,000; No payout (collect premium): +$1000.
3. Determine the probability of each outcome: Accident probability: 0.1% = 0.001; No accident: 1 – 0.001 = 0.999.
4. Multiply outcome values by their probabilities: Paying claim: -$499,000 * 0.001 = -$499; No payout: $1000 * 0.999 = $999.
5. Sum the results: -$499 + $999 = $500.

The expected value for the insurance company is $500 per policy per year. This is why insurance companies profit, because on average they can expect to make $500 from every policy they sell. However, it’s also worth noting, that, they are not guaranteed to make this amount on any single policy. They may pay out big claims, but as long as the probability calculations and pricing are done well, they can expect to profit in the long run. From a consumer perspective, it’s generally understood that buying insurance is a negative expected value situation as a cost, but most people prefer the protection and lower variance of outcomes, to the alternative.

Important Considerations When Using Expected Value

While expected value is a valuable tool, it’s essential to understand its limitations and use it responsibly:

• Long-Run Perspective: Expected value is most meaningful when considering long-run outcomes. In a single event, actual outcomes may vary significantly from the expected value. It is less relevant if a situation only occurs once.
• Risk Tolerance: Expected value does not consider risk aversion. A decision with a high expected value might also have a high potential for substantial losses that an individual might be unwilling or unable to absorb. For example a high EV bet with the risk of going bankrupt might be a bad bet to make for most people despite the EV, while the same bet would be fine for a very large corporation with large reserves.
• Accuracy of Probabilities: The accuracy of your expected value calculation depends entirely on the accuracy of the probabilities you are using. If you are using an inaccurate estimate of probability, the expected value will be equally inaccurate. Subjectivity can often creep into the probability assessment, so you should always be mindful of the source of the probabilities you use.
• Not a Guarantee: Expected value does not guarantee a specific outcome. It is an average over many repetitions. For a single occurrence, outcomes can, and frequently will, vary significantly from the expected value.
• Uncertainty of Outcomes: When analyzing real world scenarios, such as business decisions, the full range of possible outcomes, and their potential values may be difficult to define with full certainty. A sensitivity analysis of your inputs and outcomes may be advisable to validate your findings.
• Non-Monetary Outcomes: Expected value can be applied beyond monetary gains and losses. Outcomes can involve qualitative factors like time saved, improvement in performance, or satisfaction, these can be included if you can put a numerical value to them.

Practical Applications of Expected Value

The concept of expected value is far-reaching and has applications in many different fields:

• Gambling: Determining the fairness of games and long-term odds. Most casino games are designed such that the expected value for the player is negative, which ensures that the house can stay in business long term.
• Investing: Evaluating investment options by quantifying risks and potential returns. Expected value is a critical tool to assess risk versus reward.
• Insurance: Assessing the risk and pricing policies (from the insurer’s perspective). Insurers utilize large databases of historical information to assess the probability of an event, and price their policies appropriately to ensure a positive expected value for them.
• Business Decisions: Evaluating the potential outcomes of new product launches, marketing campaigns, or strategic investments. Business leaders can make better choices when analyzing expected values.
• Project Management: Analyzing the potential time and cost overruns for projects. By looking at the probability of various delays or cost increases, you can use expected value to estimate the cost and time more accurately.
• Healthcare: Assessing the risks and benefits of medical treatments and procedures. A doctor, for example, will choose treatments with the highest expected benefit based on the probability of each potential outcome.
• Everyday Decision Making: Making more informed decisions about everyday risks and opportunities, even if in a non-formal manner. This allows you to weigh potential risks and rewards.

Conclusion

Understanding and calculating expected value is a powerful tool that can improve decision-making in various situations. By following the simple steps outlined in this article and carefully considering the probabilities and outcomes, you can make more informed choices, assess risks more accurately, and understand the long-term implications of your actions. While the expected value does not guarantee specific results in any single instance, it gives you the average outcome over many plays or events and allows for decisions based on a solid quantitative basis. Mastering expected value is a crucial step in becoming a more rational and effective decision-maker in both your personal and professional life. By leveraging this concept, you can navigate uncertainty and make choices that align with your goals and risk tolerance. Remember to always consider the context of your analysis, be precise with your probabilities, and use expected value as one tool in your overall decision-making framework.