๐ What is Expected Value?
Expected Value (EV), sometimes called expectation, is a concept used in probability, statistics, and decision theory to predict the average outcome of a scenario where multiple outcomes are possible, and each outcome has an associated probability. Think of it as the long-run average result if you were to repeat the experiment or decision many times.
๐ A Little History
The concept of expected value dates back to the 17th century. It arose from the study of games of chance. Prominent mathematicians like Blaise Pascal and Pierre de Fermat were instrumental in developing the underlying theory while trying to solve problems related to gambling.
๐ Key Principles
- ๐ข Outcome Values: Each possible outcome must be assigned a numerical value representing its worth (gain or loss).
- ๐ Probabilities: Every outcome needs an associated probability of occurring. The sum of all probabilities must equal 1.
- โ Weighted Average: EV is calculated as the sum of each outcome's value multiplied by its probability.
๐งฎ The Formula
The formula for the expected value of a discrete random variable $X$ is:
$E(X) = \sum_{i=1}^{n} x_i * P(x_i)$
Where:
- $E(X)$ is the expected value of the random variable $X$.
- $x_i$ is the value of the $i$-th outcome.
- $P(x_i)$ is the probability of the $i$-th outcome.
- $n$ is the number of possible outcomes.
๐ฒ Real-World Examples
Example 1: Coin Toss Game
Suppose you're playing a game where you flip a fair coin. If it lands on heads, you win $10. If it lands on tails, you lose $5. What is the expected value of playing this game?
Solution:
- ๐ฐ Outcome 1: Heads (win $10) with probability 0.5
- ๐ Outcome 2: Tails (lose $5) with probability 0.5
$E(X) = (10 * 0.5) + (-5 * 0.5) = 5 - 2.5 = 2.5$
The expected value is $2.5. This means that, on average, you would expect to win $2.50 each time you play the game.
Example 2: Lottery Ticket
Imagine you buy a lottery ticket for $1. There is a 1 in 1,000 chance of winning $500, and a 999 in 1,000 chance of winning nothing. What is the expected value of buying a lottery ticket?
Solution:
- ๐ Outcome 1: Win $500 (net gain of $499 after deducting the $1 cost) with probability 0.001
- โ Outcome 2: Win $0 (lose $1) with probability 0.999
$E(X) = (499 * 0.001) + (-1 * 0.999) = 0.499 - 0.999 = -0.50$
The expected value is -$0.50. This means that, on average, you would expect to lose $0.50 for each lottery ticket you buy. It's important to remember that this is just an average over many trials, and you could still win the lottery!
Example 3: Investment Decision
You're considering investing in a risky stock. There is a 40% chance the stock will increase in value by 20%, and a 60% chance it will decrease in value by 10%. What is the expected return on your investment?
Solution:
- ๐ Outcome 1: Increase by 20% (0.20) with probability 0.4
- ๐ Outcome 2: Decrease by 10% (-0.10) with probability 0.6
$E(X) = (0.20 * 0.4) + (-0.10 * 0.6) = 0.08 - 0.06 = 0.02$
The expected return is 2%. This suggests that, on average, you would expect your investment to increase by 2%.
๐ก Conclusion
Understanding expected value is crucial for making informed decisions in situations involving uncertainty. While it doesn't predict the outcome of a single event, it provides a valuable tool for assessing the long-term average outcome and making rational choices. Whether you're playing games, making investments, or analyzing business strategies, expected value helps you navigate risk and maximize your potential gains.