University:
McMaster University
Course:
STATS 1LL3 | Introduction to Probability and Statistics
Academic year:

2023
Views:

214

Pages:

2
Author:

Sabrina S.

Bayes' Theorem For any two events A and B, there are two conditional probabilities that can be considered: Pr(B|A) = Pr(A ∩ B) / Pr(A) Pr(A|B) = Pr(A ∩ B) / Pr(B) If A and B are two events, then: Pr(A|B) = Pr(B|A) * Pr(A) / (Pr(B|A) * Pr(A) + Pr(B|A') * Pr(A')) Proof: Recall the definition of conditional probability: Pr(A|B) = Pr(A ∩ B) / Pr(B) And theorem 2: Pr(B) = Pr(A ∩ B) + Pr(A' ∩ B) Substituting, we have: Pr(A|B) = Pr(A ∩ B) / (Pr(A ∩ B) + Pr(A' ∩ B)) We note that: Pr(A ∩ B) = Pr(B|A) * Pr(A) Pr(A' ∩ B) = Pr(B|A') * Pr(A') Therefore: Pr(A|B) = (Pr(B|A) * Pr(A)) / (Pr(B|A) * Pr(A) + Pr(B|A') * Pr(A')) Independent Events The intuitive feeling is that independent events have no effect upon each other. But how do we decide whether two events A and B are independent? If the occurrence of event A has nothing to do with the occurrence of event B, then we expect the conditional probability of B given A to be the same as the unconditional probability of B: Pr(B|A) = Pr(B) The information that event A has occurred does not change the probability of B occurring. If Pr(B|A) = Pr(B), then, using the definition of conditional probability, Pr(B) = Pr(B ∩ A) / Pr(A) or Pr(A ∩ B) = Pr(A) × Pr(B) if events are independent In words, the probability of the intersection of independent events is the product of their individual probabilities.