Study Notes on Expected Value and Decision-Making

Expected Value (EV)

• Definition of Expected Value (EV):

  • The expected value is a key concept in statistics and probability, representing the average outcome of a random variable.

  • Mathematically defined as:
    $Ev(X) = ext{P}<em>1 x</em>1 + ext{P}<em>2 x</em>2 + ext{P}<em>3 x</em>3 + … + ext{P}<em>n x</em>n$

  • Where (Pi) represents the probability of outcome (xi).

• General Formula:

  • More formally, the expected value of a discrete random variable can be expressed as:
    Ev(X) = rac{1}{n} imes ext{sum}igg[P(xi) imes xiigg]

  • Where (P(xi)) is the probability of outcome (xi).

• Application of EV:

  • In decision-making contexts such as gambling or investment, the expected value helps evaluate the potential outcomes of different choices.

Example with Values

• Win/Lose Scenario:

  • Win Value:

  • Value = 300

  • Lose Value:

  • Value = -100

  • Probabilities:

  • Probability of winning = 20%

  • Probability of losing = 80%

• Expected Utility (EU):

  • The expected utility can be calculated similarly to expected value.

  • In this case, consider two choices based on given probabilities:

Choices Context

• Choice A:

  • Guaranteed value of 70k (where k is a thousand). Hence,

  • $EU(A) = 70,000 imes 1 = 70,000$

• Choice B:

  • Probability scenario of 70% chance at 100k and a 30% chance at 0k. Hence,

  • Calculation for expected value of choice B:

    • $EV(B) = 100,000 imes 0.7 + 0 imes 0.3$

    • Which simplifies to:

    • $EV(B) = 70,000$

• Comparative Analysis:

  • Comparing both options, both hold the same expected value of 70k.

  • However, variance and risk profiles differ between the two choices, with Choice A being risk-free and Choice B possessing an element of uncertainty.