3. Conditional probability I (annotated and updated)

ECE 226: Conditional Probability I

  • Instructor: Guosong Yang

  • Semester: Spring 2025

  • Department: Electrical and Computer Engineering, Rutgers University

Outline

  • Conditional Probability

  • Independence

Conditional Probability

Key Concepts

  • Conditional probability is a measure of the probability of an event occurring given that another event has already occurred.

  • Defined for two events A and B as:

    • P(A|B) = P(A ∩ B) / P(B) (for P(B) > 0)

Example 1.15: Fair Die Roll

  • Question: What is the probability that the outcome is an even number given it was less than or equal to 3?

  • Sample Space: S = {1, 2, 3, 4, 5, 6}

  • Events:

    • A: outcome is even {2, 4, 6}

    • B: outcome is less than or equal to 3 {1, 2, 3}

  • Therefore, P(A|B) = P(A ∩ B) / P(B) = |A ∩ B| / |B| = 1/3.

Definition of Conditional Probability

  • For events A and B:

    • P(A|B) = P(A ∩ B) / P(B)

  • It is undefined when P(B) = 0.

Properties of Conditional Probability

  • Conditional probability adheres to the axioms of probability:

    1. P(A) ≥ 0 for any event A.

    2. P(S) = 1 for a sample space S.

    3. For disjoint events A1, A2, A3,...: P(A1 ∪ A2 ∪ A3 ∪ ...) = P(A1) + P(A2) + P(A3) + ...

  • Given event B with P(B) > 0, the following holds true:

    1. P(A|B) ≥ 0 for any event A.

    2. P(B|B) = 1.

    3. For disjoint events: P(A1 | B) + P(A2 | B) + P(A3 | B) + ... = P(A1 ∪ A2 ∪ A3 | B)

Example 1.17: Fair Die Roll Twice

  • Roll a fair die twice; let the results be X1 and X2.

  • Question: Given that X1 + X2 = 7, what is the probability that X1 = 4 or X2 = 4?

  • The calculation involves conditional analysis based on the possible combinations yielding a sum of 7.

Chain Rule for Conditional Probability

  • For two events:

    • P(A ∩ B) = P(A) P(B|A) = P(B) P(A|B)

  • For three events:

    • P(A ∩ B ∩ C) = P(A) P(B|A) P(C|A,B)

  • For n events:

    • P(A1 ∩ A2 ∩ ... ∩ An) = P(A1) P(A2|A1) ... P(An|A1,A2,...,An-1)

Independence

Key Concepts

  • Definition: Events A and B are independent if:

    • P(A ∩ B) = P(A) P(B)

Example 1.20: Random Number Selection

  • Pick random number N from {1, 2, 3,..., 10}.

  • Let A = event that N < 7, B = event that N is an even number.

  • Evaluate independence:

    • P(A) = 0.6, P(B) = 0.5, check against P(A ∩ B).

Independence vs. Disjoint Events

  • Disjoint (mutually exclusive) events:

    • P(A ∩ B) = 0 implies they cannot both occur simultaneously.

  • Independent events:

    • P(A|B) = P(A) implies that the occurrence of B does not affect the probability of A occurring.

Lemma 1.2

  • Consider two events A and B with P(A) > 0 and P(B) > 0.

  • If A and B are disjoint, then they are not independent.

Reading Materials

  • Reference: [PN14, Sec. 1.4.0–1.4.1]

  • Next topics: [PN14, Sec. 1.4.2–1.4.4, Ch. 2]