probability algebra

Question 1

conditional probability

Answer 1

P(A|B) = P(ANB) / P(B)
* = P(A) if events A and B are independent
* (“N” is set-wise intersection (“and”))

Question 2

event intersections (telescoping), and independent events

Answer 2

• P(A1 N … N An) = P(A1) P(A1|A2) P(A3|A1NA2) … P(An|A1N…NA_{n-1})
• = P(A1)…P(An) if events are independent

note, P(A1,A2) is shorthand for P(A1NA2) (etc.), and where “N” is set-wise intersection (“and”)

Question 3

law of total probability

Answer 3

• let events A1,…,An partition the sample space
• let event B be some event in the sample space
• P(B) = P(A1)P(B|A1)+…+P(An)P(B|An)

Question 4

Bayes theorem

Answer 4

• let events A1,…,An partition the sample space
• the posterior probabilities of Ai conditional on event B (new information) is
  
  • P(Ai|B) = P(Ai)P(B|Ai) / sum_j P(Aj)P(B|Aj)
  • note the denominator is aka P(B)