Expected Value Notes

Expected Value

Introduction

• Expected value is a concept in probability that helps in logical thinking.

Basic Probability Experiment: Rolling a Die

• Sample space: $S = {1, 2, 3, 4, 5, 6}$
• Each outcome is equally likely.
• Probability of rolling any specific number (e.g., 3 or 6): $\frac{1}{6}$
• For each outcome $x$, there is a probability associated with $x$.
• The sum of all probabilities equals 1.

Calculating Expected Value

• Expected value is calculated by taking each outcome, multiplying it by its probability, and then summing the results.
• Formula for expected value: $E(X) = \sum x Imes P(x)$
• Example: Rolling a single die:
  • Expected value = (1 \tImes \frac{1}{6}) + (2 \tImes \frac{1}{6}) + (3 \tImes \frac{1}{6}) + (4 \tImes \frac{1}{6}) + (5 \tImes \frac{1}{6}) + (6 \tImes \frac{1}{6})
  • $= \frac{1}{6} + \frac{2}{6} + \frac{3}{6} + \frac{4}{6} + \frac{5}{6} + \frac{6}{6} = \frac{21}{6} = \frac{7}{2} = 3.5$
• The expected value for rolling a die is 3.5.

Interpretation of Expected Value

• Expected value is the average value you would expect if you repeated the experiment many times.
• It is not necessarily a value you would expect to occur in a single trial.
• For rolling a die, you won't roll a 3.5, but the average of many rolls will approach 3.5.

Definition of Expected Value

• If $A$ is an event with numerical outcomes $A<em>1, A</em>2, A<em>3, …$ and probabilities $P</em>1, P<em>2, P</em>3, …$, then the expected value of $A$ is:
  • E(A) = A1 \tImes P1 + A2 \tImes P2 + A3 \tImes P3 + …

Using Expected Value to Think Logically: A Game Example

• Game: Pay $1 to roll a die. If the number is a multiple of 3, you get $5 back.
• Possible outcomes:
  • Profit of $4 (rolling a 3 or 6).
  • Loss of $1 (rolling 1, 2, 4, or 5).
• Probabilities:
  • Probability of rolling a multiple of 3: $\frac{2}{6} = \frac{1}{3}$
  • Probability of not rolling a multiple of 3: $\frac{4}{6} = \frac{2}{3}$
• Check: Probabilities must add up to 1: $\frac{1}{3} + \frac{2}{3} = 1$
• Expected value of the game:
  • E = (4 \tImes \frac{1}{3}) + (-1 \tImes \frac{2}{3}) = \frac{4}{3} - \frac{2}{3} = \frac{2}{3}
• The expected value is $\frac{2}{3}$, meaning you would expect to make about $0.67 on average per game.

Fair Game

• A fair game has an expected value of zero.
• In a fair game, neither player has an advantage on average.
• If the expected value is negative, it's a game you shouldn't play because you will lose money on average.

Applications of Expected Value

• Investments: Evaluating the potential gains and losses of an investment.
  • Consider probabilities of a stock going up or down.
  • Calculate expected value to determine if the investment has a positive expected outcome.
• Life Insurance: Actuaries use expected value to determine insurance premiums.
  • Life insurance is a gamble where you bet you will die.
  • The insurance company bets you will live.
  • Actuaries calculate the probability of a person living a certain number of years to determine how much to charge for a policy.
  • If a 50-year-old buys a life insurance policy, the company calculates the probability of the person living 20 more years.

Conclusion

• Expected value is a simple but powerful tool for making decisions under uncertainty.
• It is calculated by multiplying each outcome by its probability and summing the results.
• It represents the average payoff you can expect over many trials.