Probability: Independence, Conditional Probability, and Related Concepts (Notes)

Flashcards covering the core concepts of independence, mutual vs pairwise independence, examples, and conditional probability as presented in the lecture notes.

What does it mean for two events A and B to be independent?

P(A|B) = P(A) (knowledge about B does not affect belief about A); equivalently P(B|A) = P(B) or P(AB) = P(A)P(B) under appropriate nonzero probabilities.

What are the three equivalent identities that characterize independence of A and B (assuming positive probabilities)?

P(A|B) = P(A); P(B|A) = P(B); P(AB) = P(A)P(B) (with P(A) > 0 and P(B) > 0 for the equivalence to hold).

In a standard deck of 52 cards, are the events 'the card is Hearts' and 'the card is a Queen' independent? Provide the probabilities.

Yes. P(H) = 13/52 = 1/4, P(Q) = 4/52 = 1/13, P(HQ) = 1/52. Since P(H)P(Q) = (1/4)(1/13) = 1/52 = P(HQ), they are independent.

Does independence imply disjointness?

No. Independence implies AB cannot be empty if P(A) > 0 and P(B) > 0; they can be non-disjoint only when at least one has probability 0.

What is the definition of mutual (or multiple) independence for events A1,…,An?

For any subcollection of events, P(intersection of the selected events) equals the product of their individual probabilities. This must hold for every subcollection.

Is mutual independence stronger than pairwise independence?

Yes. Pairwise independence requires independence for every pair, while mutual independence requires the intersection property to hold for every subcollection, not just pairs.

In the tetrahedral die example, are the events R (red on the down face) and G (green on the down face) independent? What about R and B?

Yes, they are independent (pairwise). Each has P(R)=P(G)=P(B)=1/2 and P(R∩G)=1/4, etc.

Are R, G, and B mutually independent in the tetrahedral example?

No. While they are pairwise independent, P(RGB) ≠ P(R)P(G)P(B); thus they are not mutually independent. This shows independence is stronger than pairwise independence.

What does the tetrahedral example illustrate about independence vs pairwise independence?

Independence is strictly stronger than pairwise independence; pairwise independence does not guarantee mutual independence.

What is P(E|F) in terms of P?

P(E|F) = P(E ∩ F) / P(F), provided P(F) ≠ 0.

What is πF and what are its basic probability properties when defined as πF(E) = P(E|F)?

πF is a probability measure on the same event space defined by conditional probability. Properties: 0 ≤ πF(E) ≤ 1; πF(S) = 1; and for any countable collection of disjoint events Ei, πF(∪ Ei) = Σ πF(E_i).

How is independence of A and B expressed in terms of joint probability P(AB)?

A and B are independent if P(AB) = P(A)P(B).

What is the difference between pairwise independence and mutual independence?

Pairwise independence means every pair of events is independent; mutual independence means all subcollections of events are independent (the strongest form).

What is the definition of conditional probability P(E|F) in general terms?

P(E|F) = P(E ∩ F) / P(F) provided P(F) ≠ 0.

What are the axioms of a probability function P and how do they relate to conditional probability πF?

Axiom 1: P(A) ≥ 0 for all events A. Axiom 2: P(S) = 1. Axiom 3: countable additivity for disjoint events. Conditional probability πF defined by πF(E) = P(E|F) satisfies 0 ≤ πF(E) ≤ 1, πF(S) = 1, and similar additivity properties for disjoint E_i.