Law of Total probability and Bayes' theorem

Question 1

Simple law of total probability - 2

Answer 1

• A and B are two events

- P(B) = P ( B ∩ A ) + P ( B ∩ A bar ) = P ( B | A ) · P ( A ) + P ( B | A bar ) · P ( A bar )

Question 2

Bayes’ Theorem - 3

Answer 2

• P(A|B)= P(B|A)·P(A) / P(B|A)·P(A)+P(B|A bar)·P(A bar).
• P(B|A) + P(B bar | A) = 1
• P(B|A) + P(B|A bar ) ≠ 1

Question 3

Partition - 3

Answer 3

• A collection of A1, … , Am events
• Each event is pairwise disjoint (Ai n Aj) = Empty if i ≠ j
• The union of events is the entire sample space

Question 4

Law of total probability - 2

Answer 4

• A1, … , Am is a partition

- P(B) = Sum(m, i = 1) P(B ∩ Ai ) = Sum(m, i = 1) P(B|Ai ) · P(Ai ).