Conditional Probability

1. 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 week 3

2. 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: week 3

3. The fact that conditional probability is a valid probability measure allows the following: • , A, B F, P(A) >0 for any A, B1, B2 F, P(A) >0. week 3

4. 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: week 3

5. 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: week 3

6. Examples • 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? week 3

7. Bayes’ Rule • Let be a partition of Ω such that for all i then for any . • 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 week 3

8. 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. week 3

9. 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. week 3

10. Example • Roll a die twice. Define the following events; A: 1st die odd B: 2nd die odd C: sum is odd. week 3

11. Example • Let R, S and T be independent, equally likely events with common probability 1/3. What is ? • Solution: week 3

12. Claim • If A, B are independent so are and and . • Proof: week 3