# Expected value

Expected value is a statistical concept that calculates the average outcome of a random variable over numerous trials, used to predict the long-term results of various scenarios in finance, economics, and decision-making.
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Updated: Jun 13, 2024

## 3 key takeaways:

• Expected value is the average result of a random variable if the process is repeated many times.
• It is calculated by multiplying each possible outcome by its probability and summing these products.
• Expected value helps in decision-making by providing a measure of the central tendency of uncertain outcomes.

## What is expected value?

Expected value (EV) is a fundamental concept in probability and statistics that represents the average or mean value of a random variable over a large number of trials or experiments. It provides a single summary measure that captures the central tendency of the variable’s possible outcomes. In finance and economics, expected value is used to make informed decisions by evaluating the potential outcomes of uncertain events and their associated probabilities.

## How is expected value calculated?

The expected value of a random variable is calculated by multiplying each possible outcome by its probability and then summing these products. The formula for expected value (EV) is:

[ \text{EV} = \sum (x_i \cdot p_i) ]

where:

• ( x_i ) represents each possible outcome,
• ( p_i ) represents the probability of each outcome.

For example, if you have a simple lottery ticket with a 50% chance of winning $100 and a 50% chance of winning nothing, the expected value of the lottery ticket is: [ \text{EV} = (100 \cdot 0.5) + (0 \cdot 0.5) = 50 ] This means that, on average, you can expect to win$50 per ticket over many trials.

## Key features of expected value:

Expected value has several important features and implications:

• Long-Term Average: Expected value represents the average outcome over a large number of trials, providing a measure of central tendency for a random variable.
• Decision-Making Tool: It helps in making decisions under uncertainty by comparing the expected values of different options. Higher expected value is often preferred as it indicates a more favorable average outcome.
• Risk Assessment: By considering the probabilities and outcomes, expected value helps assess the potential risk and return of different scenarios, particularly in finance and investment.
• Fairness and Pricing: In gambling and insurance, expected value is used to determine fair pricing and to ensure that, over the long run, neither party (gambler or insurer) systematically wins or loses.

## Applications of expected value:

Expected value is widely used in various fields, including:

1. Finance and Investment: Investors use expected value to evaluate the potential returns of different investments, considering the probability of various outcomes and their associated returns.
2. Economics: Economists apply expected value to model uncertain economic scenarios and to make policy recommendations based on the average expected outcomes.
3. Gambling and Games of Chance: Expected value helps gamblers and game designers determine the fairness of bets and games, ensuring that the long-term outcomes are balanced.
4. Insurance: Insurers use expected value to calculate premiums by assessing the average expected loss from different risks, ensuring that the premiums cover the expected payouts.
• Probability theory: Understanding the mathematical foundation for calculating expected value and other statistical measures.
• Risk and return: Insights into how expected value is used to balance potential risks and rewards in investment decisions.
• Decision theory: Exploring the principles and models used to make rational decisions under uncertainty, often using expected value as a key metric.

Exploring these related topics will provide a comprehensive understanding of expected value, its calculation, and its significance in various applications of decision-making under uncertainty.

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Arti is a specialized AI Financial Assistant at Invezz, created to support the editorial team. He leverages both AI and the Invezz.com knowledge base, understands over 100,000... read more.