Week 2 notes

Question 1

How is conditional probability defined?

Answer 1

For events A and B, with P(B) > 0, the conditional probability of A given B, denoted P(A | B), is given by:
P(A | B) = P(A intersect B)/P(B)

Question 2

What is the multiplication law?

Answer 2

P(A intersect B) = P(A | B)P(B) = P(B | A)P(A)

Question 3

What is the law of total probability?

Answer 3

Suppose events B1, B2, … , Bn satisfy the following:

i) S = B1 U B2 U … U Bn (S is partitioned by the events)
ii) Bi intersect Bj = the empty set for every i is not equal to j
iii) P(Bi) > 0, for i = 1,…,n

We say that the events B1, … ,Bn partition the sample space S. For any event A, we can write
P(A) = P(A | B1)P(B1) + … + P(A | Bn)P(Bn)

Question 4

What is Bayes rule?

Answer 4

Let B1, … , B2 partition S and let A be an event with
P(A) > 0. Then, for j=1, … , n, we have
P(Bj | A) = (P(A | Bj)*(P(Bj))/P(A)

Question 5

When are events A and B independent?

Answer 5

Events A and B are independent if knowing that one occurs does not affect the probability that the other occurs , i.e.
P(A | B) = P(A) and P(B | A) = P(B)

Question 6

What is another way to define the independence of events A and B?

Answer 6

Events A and B are independent if

P(A intersect B) = P(A)*P(B)

Question 7

How is independence defined for three or more events?

Answer 7

Let A1, A2, … , Ak, where k >=3, be a collection of events.
1: The events are pairwise independent if every pair of events in the collection are independent
2: The events are mutually independent if every subcollection of events, say Ai1, Ai2, … , Ain, satisfy the following:
P(Ai1 intersect Ai2 intersect … intersect Ain =
P(Ai1) x P(Ai2) x P(Ain)

Question 8

If a collection of sets are pairwise independent, does that imply that the collection of sets is also mutually independent?

Answer 8

No