1. week 21 Rules of Probability for all Corollary: The probability of the union of any two events A and B is Proof: … If then, Proof:

2. week 22 Inclusion / Exclusion formula: For any finite collection of events Proof: By induction

3. week 23 Example In a lottery there are 10 tickets numbered 1, 2, 3, …, 10. Two numbers are drown for prizes. You hold tickets 1 and 2. What is the probability that you win at least one prize?

4. week 24 Conditional Probability Idea – have performed a chance experiment but don’t know the outcome (ω), but have some partial information (event A) about ω. Question: given this partial information what’s the probability that the outcome is in some event B? Example: Toss a coin 3 times. We are interested in event B that there are 2 or more heads. The sample space has 8 equally likely outcomes. The probability of the event B is … Suppose we know that the first coin came up H. Let A be the event the first outcome is H. Then and The conditional probability of B given A is

5. week 25 Given a probability space (Ω, F, P) and events A, B F with P(A) > 0 The conditional probability of B given the information that A has occurred is Example: We toss a die. What is the probability of observing the number 6 given that the outcome is even? Does this give rise to a valid probability measure? Theorem If A F and P(A) > 0 then (Ω, F, Q) is a probability space where Q : is defined by Q(B) = P(B | A). Proof:

6. week 26 The fact that conditional probability is a valid probability measure allows the following: , A, B F, P(A) >0  for any A, B 1, B 2 F, P(A) >0.

7. week 27 Multiplication rule For any two events A and B, For any 3 events A, B and C, In general, Example: An urn initially contains 10 balls, 3 blue and 7 white. We draw a ball and note its colure; then we replace it and add one more of the same colure. We repeat this process 3 times. What is the probability that the first 2 balls drawn are blue and the third one is white? Solution:

8. week 28 Law of total probability Definition: For a probability space (Ω, F, P), a partition of Ω is a countable collection of events such that and Theorem: If is a partition of Ω such that then for any. Proof:

9. week 29 Examples 1.Calculation of for the Urn example. 2. In a certain population 5% of the females and 8% of the males are left-handed; 48% of the population are males. What proportion of the population is left- handed? Suppose 1 person from the population is chosen at random; what is the probability that this person is left-handed?

10. week 210 Bayes’ Rule Example: A test for a disease correctly diagnoses a diseased person as having the disease with probability 0.85. The test incorrectly diagnoses someone without the disease as having the disease with probability 0.1 If 1% of the people in a population have the disease, what is the probability that a person from this population who tests positive for the disease actually has it? (a) 0.0085 (b) 0.0791 (c) 0.1075 (d) 0.1500 (e) 0.9000

11. week 211 Independence Example: Roll a 6-sided die twice. Define the following events A : 3 or less on first roll B : Sum is odd. If occurrence of one event does not affect the probability that the other occurs than A, B are independent.

12. week 212 Definition Events A and B are independent if Note: Independence ≠ disjoint. Two disjoint events are independent if and only if the probability of one of them is zero. Generalized to more than 2 events: A collection of events is (mutually) independent if for any subcollection Note: pairwize independence does not guarantee mutual independence.

13. week 213 Definition Events A and B are independent if Note: Independence ≠ disjoint. Two disjoint events are independent if and only if the probability of one of them is zero. Generalized to more than 2 events: A collection of events is (mutually) independent if for any subcollection Note: pairwize independence does not guarantee mutual independence.

14. week 214 Example Roll a die twice. Define the following events; A: 1st die odd B: 2nd die odd C: sum is odd.

15. week 215 Example

16. week 216