Probability Trees and Conditional Expectations

What is the expected value of a random variable?

The probability-weighted average of all possible outcomes. E(X) = Σ[P(Xᵢ) × Xᵢ].

What must the probabilities of mutually exclusive and exhaustive outcomes sum to?

1.00, or 100%.

Does the expected value have to be a possible realized outcome?

No. It is a probability-weighted average and may not be an actual possible outcome.

What is variance?

A measure of dispersion around the expected value. σ² = Σ[P(Xᵢ) × (Xᵢ − E(X))²].

What is standard deviation?

The square root of variance: σ = √σ². It measures dispersion in the same units as the original variable.

What is the calculation sequence for expected value, variance, and standard deviation?

E(X) → calculate deviations → square deviations → probability-weight them → sum for variance → √variance for standard deviation.

How is standard deviation interpreted in investment analysis?

Higher standard deviation means greater dispersion of possible outcomes and therefore greater uncertainty or risk, all else equal.

What is a probability tree?

A diagram showing sequential events and their unconditional and conditional probabilities.

What is the key rule for probability trees?

Multiply probabilities down a path to obtain joint probabilities; add probabilities across relevant mutually exclusive paths.

What is joint probability?

The probability that two events occur together. P(A ∩ B) = P(A) × P(B | A).

What is conditional probability?

The probability of an event given that another event has already occurred. P(A | B) means "probability of A given B."

In P(A | B), what does the event after "|" represent?

The information already known or assumed to have occurred.

What is a conditional expected value?

The expected value given that a particular state is known to have occurred: E(X | A) = Σ[P(Xᵢ | A) × Xᵢ].

How is an unconditional expected value calculated from conditional expectations?

E(X) = Σ[P(Stateᵢ) × E(X | Stateᵢ)].

What is the difference between conditional and unconditional expectation?

Conditional expectation assumes a particular state is known; unconditional expectation incorporates uncertainty over all possible states.

What are independent events?

Events where occurrence of one does not change the probability of the other: P(A | B) = P(A).

What is the joint probability rule for independent events?

P(A ∩ B) = P(A) × P(B).

What is the difference between independent and mutually exclusive events?

Independent: knowing one occurred gives no information about the other. Mutually exclusive: if one occurs, the other cannot occur.

Can two non-zero-probability mutually exclusive events be independent?

No. Mutual exclusivity means P(A ∩ B) = 0, while independence requires P(A ∩ B) = P(A)P(B) > 0.

What is Bayes' formula?

P(A | B) = [P(B | A) × P(A)] ÷ P(B).

What is the purpose of Bayes' formula?

To update the probability of an event after receiving new information.

What is the prior probability?

P(A): the probability assigned to an event before receiving new information.

What is the posterior probability?

P(A | B): the updated probability after incorporating new information.

What is the likelihood in Bayes' formula?

P(B | A): the probability of observing the information given that the event occurred.

What does P(B) represent in Bayes' formula?

The unconditional probability of observing the new information across all possible states.

What is the total probability rule used in Bayes questions?

P(B) = Σ[P(B | Aᵢ) × P(Aᵢ)]. It calculates the unconditional probability of the observed information.

What is the key Bayes intuition?

Prior belief → new information → posterior belief.

What is the most common Bayes exam mistake?

Confusing P(A | B) with P(B | A). The likelihood is not the posterior.

How can Bayes be solved using a joint-probability table?

For each state: Prior × Likelihood = Joint probability. Add the joint probabilities to get P(Information), then divide the desired joint probability by P(Information).

What is the Bayes table shortcut?

Multiply across → add down → divide.

What are diffuse priors?

Equal prior probabilities assigned to all possible states when there is no prior information favoring one state.

If there are n states with diffuse priors, what is each prior?

1 ÷ n.

How does Bayes simplify with diffuse priors?

Because all priors are equal, they cancel: P(Stateⱼ | Information) = P(Information | Stateⱼ) ÷ ΣP(Information | Stateᵢ).

What is the key intuition behind diffuse priors?

All states start equally likely, so the relative likelihood of the new information determines the posterior.

What is the frequency interpretation of Bayes?

Once information B is known, restrict the universe to observations containing B. P(A | B) = observations containing both A and B ÷ observations containing B.

What is the counting multiplication rule?

If sequential operations can occur in n₁, n₂, …, nₖ ways, the total number of possible sequences is n₁ × n₂ × … × nₖ.

What is factorial?

n! = n × (n − 1) × … × 1; 0! = 1. It counts arrangements of all n distinct objects.

What is a combination?

The number of ways to choose r objects from n when order does NOT matter: C(n,r) = n! ÷ [(n − r)!r!].

What is a permutation?

The number of ways to choose and arrange r objects from n when order DOES matter: P(n,r) = n! ÷ (n − r)!.

What is the key question for combination vs permutation?

Does order matter? No → combination. Yes → permutation.

What is a useful mnemonic for combination vs permutation?

Combination = Committee (order irrelevant). Permutation = Podium (order matters).

What is the labeling formula?

n! ÷ (n₁! × n₂! × … × nₖ!). It is used to assign n objects among multiple labeled groups of specified sizes.

What are the main exam traps in this module?

Reversing P(A | B) and P(B | A); confusing likelihood with posterior; adding instead of multiplying down a probability tree; confusing independence with mutual exclusivity; and confusing combinations with permutations.