definition of probability (week 2)

Question 1

Probabilities as a function with some properties?

Answer 1

Axiom 1. P is a function of A : A c S to [0,1]
Axiom 2. P(∅) = 0 , P(S) = 1
Axiom 3. If A1, A2, … are disjoint events
P U limit infinity start i = 1 = sigma limit infinity start i P(Ai)

Question 2

Classical vs Bayesian

Answer 2

Classical : long-run frequency of that event occurring if the experiment is run many many times. want to get the best estimate of the true long-run probability

Bayesian: event is the subjective belief about the chance of that event occurring. want to update their belief about the probability of A in a scientific way by incorporating the infromation in the observed data

Question 3

Conditional probabilites

Answer 3

P(A|B) = P(AnB)/P(B)

Question 4

Bayes rule P(A|B) ?

Answer 4

[P(B|A)P(A)]/P(B)

Question 5

Law of Total Probability

Answer 5

P(B) = P(BnA) + P(BnA^c) = P(B|A) P(A) + P(B|A^c)P(A^c)

Question 6

Independent if P(A|B) ?

Answer 6

equals to P(A)

Question 7

if independent, P(AnB) ?

Answer 7

P(A) x P(B)

Question 8

What if prob A given B and C?

Answer 8

P(AnBnC) / P(BnC)