🟣 Expected Value

Expected Value is a way to calculate the average outcome you can expect from a probabilistic situation over the long run. It is particularly useful in decision-making under uncertainty, such as in games of chance, investments, and business decisions. Expected value helps us to determine what is "fair" in a game, or to compare different options in terms of their average payoff.

Expected Value (Expectation) of a random variable is the average outcome of an experiment you would expect to see over many repetitions. It is calculated by summing up the product of each possible outcome and its probability.

To calculate the expected value of a random variable, we need to know all possible outcomes and their associated probabilities. If we let $X$ be a random variable representing the outcomes of an experiment, and $x_1, x_2, \ldots, x_n$ be the possible values of $X$ with corresponding probabilities $P(X=x_1), P(X=x_2), \ldots, P(X=x_n)$, then the expected value of $X$, denoted as $E(X)$ or $\mu$, is given by:

$$E(X) = x_1 \cdot P(X=x_1) + x_2 \cdot P(X=x_2) + \ldots + x_n \cdot P(X=x_n)$$

In more compact summation notation, this is:

$$E(X) = \sum_{i=1}^{n} x_i \cdot P(X=x_i)$$

Where:

• $E(X)$ is the expected value of the random variable $X$
• $x_1, x_2, \ldots, x_n$ are the possible outcomes
• $P(X=x_i)$ is the probability of each outcome $x_i$
• $n$ is the total number of possible outcomes

The expected value is a weighted average of the possible outcomes, where each outcome is weighted by its probability.

Example 1: Expected Value of a Die Roll

What is the expected value when you roll a fair six-sided die?

Solution:

• Possible outcomes when rolling a die: $\{1, 2, 3, 4, 5, 6\}$
• Since the die is fair, each outcome has a probability of $\frac{1}{6}$

Using the expected value formula:

$$E(X) = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6}$$

$$E(X) = \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5$$

Answer: The expected value of rolling a fair six-sided die is 3.5. Note that 3.5 is not a possible outcome itself, but it represents the average value you would expect to get over many rolls.

Example 2: Expected Value in a Simple Lottery

Consider a lottery where you pick one number from 1 to 10. If your number matches the winning number, you win $10. If it doesn't, you lose $1. What is the expected value of playing this lottery once?

Solution:

Possible outcomes and payoffs:

• Win (number matches): Payoff = +$10. Probability of winning = $\frac{1}{10}$
• Lose (number doesn't match): Payoff = -$1 (cost of playing). Probability of losing = $\frac{9}{10}$

Using the expected value formula with payoffs as outcomes:

$$E(\text{Lottery}) = (\$10) \cdot \frac{1}{10} + (-\$1) \cdot \frac{9}{10}$$

$$E(\text{Lottery}) = \$1 - \$0.9 = \$0.10$$

Answer: The expected value of playing this lottery once is $0.10 or 10 cents. This means that on average, for each ticket you buy, you can expect to gain 10 cents in the long run. A positive expected value suggests the game is favorable to the player in the long term.

Example 3: Expected Value of Multiple Dice

What is the expected value when rolling two fair six-sided dice and summing their values?

Solution:

Using the linearity of expectation: $E(X + Y) = E(X) + E(Y)$

From Example 1, we know $E(\text{one die}) = 3.5$

Therefore:

$$E(\text{two dice sum}) = E(\text{die 1}) + E(\text{die 2}) = 3.5 + 3.5 = 7$$

Answer: The expected sum of two dice is 7.

In the context of games of chance, the concept of expected value is closely tied to the idea of a fair game.

A Fair Game is a game where the expected value is zero. In a fair game, in the long run, neither the player nor the game organizer is expected to profit or lose on average.

If the expected value of a game is positive for a player, it means the game is favorable to the player in the long run (on average, the player gains). If the expected value is negative, the game is unfavorable to the player (on average, the player loses), and favorable to the game organizer.

Example 4: Is the Lottery a Fair Game?

In Example 2, the expected value of the lottery was $0.10 (10 cents). Is this lottery a fair game?

Solution:

No, this lottery is NOT a fair game because the expected value is not zero. Since the expected value is positive ($0.10), the game is actually favorable to the player. In reality, most lotteries have negative expected values for players, making them favorable to the organizers.

Example 5: Making a Fair Coin Game

You're designing a coin flip game. You flip a fair coin. If it lands on heads, you win a prize. If it lands on tails, you pay a penalty. If the prize is $20, what should the penalty be to make this a fair game?

Solution:

For a fair game, $E(X) = 0$

Let penalty = $p$ (negative value)

$E(X) = (\$20) \cdot \frac{1}{2} + p \cdot \frac{1}{2} = 0$

$\$10 + \frac{p}{2} = 0$

$\frac{p}{2} = -\$10$

$p = -\$20$

Answer: The penalty should be $20 to make this a fair game.

Expected value is a powerful tool for comparing different options and making rational decisions when faced with uncertainty. In general, when choosing between options, selecting the one with the highest expected value maximizes your average gain over the long run.

Example 6: Investment Decision

You have two investment options:

• Option A: 60% chance of gaining $1000, 40% chance of losing $500
• Option B: 50% chance of gaining $800, 50% chance of gaining $200

Which option has the higher expected value?

Solution:

Option A: $E(A) = (\$1000)(0.6) + (-\$500)(0.4) = \$600 - \$200 = \$400$

Option B: $E(B) = (\$800)(0.5) + (\$200)(0.5) = \$400 + \$100 = \$500$

Answer: Option B has the higher expected value ($500 vs $400), so it's the better choice based on expected value alone.

1️⃣ Insurance Pricing

Insurance companies use expected value to price premiums. They calculate the expected payout (probability of claim × claim amount) and set premiums above that to ensure profit.

2️⃣ Business Decisions

Companies evaluate projects by calculating expected returns, considering probabilities of success and failure.

3️⃣ Casino Games

All casino games are designed with negative expected values for players (positive "house edge" for casino), ensuring long-term profits for the casino.

4️⃣ Medical Treatment Decisions

Doctors consider expected outcomes when recommending treatments, weighing success probabilities against risks.

5️⃣ Stock Market Investing

Investors analyze expected returns based on historical data and market forecasts to make portfolio decisions.