University:
Massachusetts Institute of Technology
Course:
6.041SC | Probabilistic Systems Analysis and Applied Probability
Academic year:

2024
Views:

444

Pages:

2
Author:

RonoEnduh6sl

LECTURE 2 Review of probability models • Readings: Sections 1.3-1.4 • Sample space Ω – Mutually exclusive Collectively exhaustive Lecture outline – Right granularity • Review • Event: Subset of the sample space • Conditional probability • Allocation of probabilities to events 1. P(A) ≥ 0 2. P(Ω) = 1 • Three important tools: – Multiplication rule 3. If A ∩ B = Ø, then P(A ∪ B) = P(A) + P(B) – Total probability theorem – Bayes’ rule 3’. If A1, A2, . . . are disjoint events, then: P(A1 ∪ A2 ∪ · · · ) = P(A1) + P(A2) + · · · • Problem solving: – Specify sample space – Define probability law – Identify event of interest – Calculate... Conditional probability Die roll example 4 A Y = Second 3 B roll 2 1 • P(A | B) = probability of A, given that B occurred 2 1 3 4 X = First roll – B is our new universe • Definition: Assuming P(B) = $ 0, • Let B be the event: min(X, Y ) = 2 P(A ∩ B) P(A | B) = P(B) • Let M = max(X, Y ) • P(M = 1 | B) = P(A | B) undefined if P(B) = 0 • P(M = 2 | B) = 1 Multiplication rule Multiplication rule Models based on conditional probabilities (A∩∩B B ∩ C) P(B | A) · P(C (C ||A A∩ ∩ B) PP (A C) = = PP(A) (A)·P A)P B) P(C | A B) A B C U P(B | A) P(Bc | A) A Bc C U A P(A) U A P(Ac)=0.95 U P(Bc | A)=0.01 Bc A Bc Cc U P(B | Ac)=0.10 P(Bc | Ac)=0.90 U P(A)=0.05 U A B P(B | A)=0.99 U U • Event A: Airplane is flying above Event B: Something registers on radar screen P(Ac) Ac P(A ∩ B) = P(B) = P(A | B) = Total probability theorem Bayes’ rule • “Prior” probabilities P(Ai) – initial “beliefs” • Divide and conquer • Partition of sample space into A1, A2, A3 • We know P(B | Ai) for each i • Have P(B | Ai), for every i A1 • Wish to compute P(Ai | B) – revise “beliefs”, given that B occurred B A1 A2 A3 A2 • One way of computing P(B): P(B) = B A3 P(A1)P(B | A1) + P(A2)P(B | A2) + P(A3)P(B | A3) P(Ai | B) = = P(Ai ∩ B) P(B) P(Ai)P(B | Ai) P(B) P(Ai)P(B | Ai) = ! j P(Aj )P(B | Aj ) 2

6.041SC Probabilistic Systems Analysis, Lecture 2 Notes: Conditioning and Bayes' Rule

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