Math Refresher for Decision Analysis

Probability Axioms and Vocabulary

  1. An outcome is something that can happen.
  2. An event is a set of one or more outcomes.
  3. The probability of an event that cannot happen is 0.
  4. The probability of an event that will certainly happen is 1.
  5. If the occurrence of one event precludes the occurrence of another, we say they are "mutually exclusive."
  6. If a set of outcomes includes every possible outcome we say it is "exhaustive."
  7. We use the word OR in its usual sense: the event "outcome A or outcome B" means that either A happens, or B happens, or both happen. The event "A or B" does not occur only when both outcome A and outcome B do not occur.
  8. We use the word AND in its usual sense: the event "outcome A and outcome B" means that both A happens and B happens. The event "A and B" does not occur if A occurs but B does not, or vice versa, or if both do not occur.

Later we will introduce the idea of independence.

Expected Value

By "expected" in the term "expected value" we mean the average outcome over the long haul. We speak generically of an "experiment" as any situation in which probabilities and unknown outcomes are involved. And so we say that the expected value of an experiment is the average outcome if the experiment were repeated many times.

Let's imagine an experiment in which we roll a single die. If it comes up 6, you win $6, otherwise nothing. If you played this game many times, what do you think your average winnings would be (in other words, your total winnings divided by the number of times you play)?

Another way to describe what we can expect to happen over many, many repetitions, the expected value, is as the weighted average of the outcomes.

\begin{equation} E(experiment) = p(x_1) v(x_1) + p(x_2) v(x_2) \end{equation}
\begin{align} E(experiment) = 0.25 x \$10 + 0.75 x -\$2 \end{align}
\begin{align} E(experiment) = \$2.50 - \$1.50 \end{align}

Problems for Practice

  1. Consider an honest six-sided die. What is the probability of rolling a 2?
  2. If you roll two such dice, what's the probability of rolling two sixes?
  3. Consider an honest coin. What are the possible outcomes of flipping it once? Twice? What's the probability that two flips will not be the same?
  4. If offered $1 or a coin flip chance at $5, which should you take? Think it through by computing, explicitly, the expected value of each option.
  5. If we play a dice game in which you receive from me as many dollars as the number on the die when the number is even and I receive the number on the die from you if the number is odd, what is the expected value of your takeaway?