Uncertainty, Expected Value, and Fair Games

Learning Objectives

• In this section, we discuss the notion of uncertainty. Mathematical preliminaries discussed here form the basis for analyzing individual decision-making in uncertain situations.
• The student should pick up the tools from this section, as they will be applied later.

As we learned in Chapter 1, "The Nature of Risk: Losses and Opportunities", and Chapter 2, "Risk Measurement and Metrics", risk and uncertainty depend on each other. The origins of this distinction go back to Mr. Knight, who distinguished between risk and uncertainty, arguing that measurable uncertainty is risk. In this section, since we focus on measurable uncertainty, we will not distinguish between risk and uncertainty, using the two terms interchangeably.

As described in Chapter 2, "Risk Measurement and Metrics", the study of uncertainty originated in games of chance. When we play games of dice, we deal with outcomes that are inherently uncertain. The science of uncertain outcomes is probability and statistics. Note that probability and statistics apply only when outcomes are uncertain. For example, a student who attends no lectures and does no work will definitely fail, while a student who scores 100% on all assignments will definitely receive an 'A'. In between these extremes lies uncertainty. Students often research instructors to estimate their likely grade.

Though we covered some probability and uncertainty concepts in Chapter 2, we repeat them here for reinforcement. Figuring out the chance of an event is the same as calculating its probability. Empirically, we repeat an experiment with uncertain outcomes and count how often the event of interest occurs. The probability is the number of occurrences divided by the total trials. For example, if you log the number of times a computer crashes daily over a year, the probability is the number of crashes divided by 365.

For some problems, the probability can be deduced mathematically, such as calculating the probability of a head in a coin toss or two aces from a deck of 52 cards. In these cases, experiments aren't needed to compute probabilities. Finally, subjective probability is based on personal beliefs and experiences, not empirical or mathematical calculations.

Consider a lottery where multiple outcomes are possible with known probabilities. Typically, monetary prizes are awarded. If a die is rolled with outcomes from $1 to $6, only one amount can be won per roll. However, if the game is played repeatedly, what is the expected amount to win?

Mathematically, the expected value (E) of the game is:

$E(U) = ∑^∞ _{i=1} π_i U(W_i) = \dfrac{1}{2} \times ln(2) + \dfrac{1}{4} \times ln(4) + … = ∑^∞ _{i=1} \dfrac{1}{2^i} ln(2^i)$

The probabilities sum to 1:

$∑^N _{i=1} π_i = 1$

The expected value of the game is calculated by multiplying each outcome by its probability and summing them. For example, in the dice game, the expected value is $\dfrac{1}{6} \times (1+2+3+4+5+6) = 3.50$. Note that this value is a long-term concept; in a single game, $3.50 will never be an actual outcome.

Key Takeaways

• Students should be able to explain probability as a measure of uncertainty in their own words.
• Students should also be able to explain that expected value is the sum of products of probabilities and outcomes and be able to compute expected values.

Discussion Questions

1. Define probability. How many ways can one estimate the probability of an event?
2. Explain the need for utility functions using the St. Petersburg paradox.
3. What number do you expect when rolling a six-faced die?
4. What is an actuarially fair game?