Expected Value Notes

Expected Value

Rolling a Single Die: Uniform PDF

  • When rolling a fair six-sided die, each outcome (1, 2, 3, 4, 5, or 6) has an equal probability of 1/6. This is a uniform probability distribution function (PDF).

Definition of Expected Value

  • The expected value represents the average outcome if an experiment is repeated many times.
  • It's calculated by multiplying each possible outcome by its probability and summing these products.

Calculation of Expected Value for Rolling a Die

  • The expected value (denoted as \mu ) for rolling a die is calculated as follows:
    \mu = (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})
  • Simplifying the equation:
    \mu = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5
  • Therefore, the expected value of rolling a die is 3.5.

Interpretation of Expected Value

  • The expected value (3.5 in the die example) is NOT a likely outcome of a single trial, but rather the average outcome over many trials.
  • The expected value can be visualized as the balancing point of the PDF. For a uniform distribution, it's the midpoint.

Non-Uniform, Non-Symmetric PDF Example

  • Consider an experiment with outcomes 1, 2, 3, 4, and 5, with probabilities 10%, 50%, 25%, 10%, and 5% respectively.

  • Calculation: \mu = (1 \times 0.10) + (2 \times 0.50) + (3 \times 0.25) + (4 \times 0.10) + (5 \times 0.05)
    \mu = 0.1 + 1.0 + 0.75 + 0.4 + 0.25 = 2.5

  • The expected value for this distribution is 2.5.

Using Spreadsheets for Expected Value Calculation

  • Organize outcomes in column A and their corresponding probabilities in column B.
  • Create a column C with the products of outcomes and probabilities (A*B).
  • Sum the values in column C to find the expected value.

Application of Expected Value in Decision Making

  • People weigh potential outcomes (good and bad) and their probabilities when making decisions.
  • Expected value helps determine if, on average, the outcome is likely to be positive or negative.
  • Example: Weighing the potential benefits and risks (probabilities) of surgery to decide whether to proceed.

Game Example: Rolling a Die with Winnings and Losses

  • A game: Pay $1 to roll a die. If a 6 is rolled, win $5; otherwise, lose the $1.
  • Outcomes: Win $4 (with probability 1/6) or lose $1 (with probability 5/6).
  • Expected Value Calculation:
    \mu = (-1 \times \frac{5}{6}) + (4 \times \frac{1}{6}) = \frac{-5}{6} + \frac{4}{6} = \frac{-1}{6} \approx -0.1667
  • Interpretation: On average, a player will lose approximately $0.1667 per game. If played 100 times, expected loss is about $16.

Fair Game

  • A fair game has an expected value of zero, meaning neither player has an advantage on average.

Pascal's Wager

  • Pascal's wager is a philosophical argument about whether to believe in God, based on the potential outcomes and their associated expected values.