30
Lecture 2: Conditional Probability Ziyu Shao School of Information Science and Technology ShanghaiTech University September 26, 2016 Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 1 / 60 Outline 1 Definition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of Events 5 Conditioning As A Problem-Solving Tool 6 Pitfalls & Paradoxes 7 Advanced Part Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 2 / 60

Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

  • Upload
    lamdat

  • View
    219

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Lecture 2: Conditional Probability

Ziyu Shao

School of Information Science and TechnologyShanghaiTech University

September 26, 2016

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 1 / 60

Outline

1 Definition & Intuition

2 Bayes’ Rule & The Law of Total Probability

3 Conditional Probabilities are Probabilities

4 Independence of Events

5 Conditioning As A Problem-Solving Tool

6 Pitfalls & Paradoxes

7 Advanced Part

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 2 / 60

Page 2: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Outline

1 Definition & Intuition

2 Bayes’ Rule & The Law of Total Probability

3 Conditional Probabilities are Probabilities

4 Independence of Events

5 Conditioning As A Problem-Solving Tool

6 Pitfalls & Paradoxes

7 Advanced Part

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 3 / 60

Thinking Conditionally

New data & information may affect our uncertainties

Conditional probability: how to update our belief?

All probabilities are conditional! (explicit/implicit background info orassumption)

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 4 / 60

Page 3: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Thinking Conditionally

Conditioning :a powerful problem-solving strategy

Reducing a complicated probability problem to a bunch of simplerconditional probability problems

First-step analysis: obtain recursive solution to multi-stage problems

Conditioning is the soul of statistics!

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 5 / 60

Definition of Conditional Probability

Definition

If Aand B are events with P(B) > 0, then the conditional probability of Agiven B, denoted by P(A|B), is defined as

P(A|B) =P(A ∩ B)

P(B).

P(A): prior probability of A.

P(A|B): posterior probability of A.

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 6 / 60

Page 4: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Example: Two Cards

A standard deck of cards is shuffled well. Two cards are drawn randomly,one at a time without replacement. Let A be the event that the first cardis a heart, and B be the event that the second card is red. Find P(A|B)and P(B|A).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 7 / 60

Example: A Girl Born in Winter

A family has two children. Find the probability that both children are girls,given that at least one of the two is a girl who was born in winter. Assumethat the four seasons are equally likely and that gender is independent ofseason (this means that knowing the gender gives no information aboutthe probabilities of the seasons, and vice versa).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 8 / 60

Page 5: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Outline

1 Definition & Intuition

2 Bayes’ Rule & The Law of Total Probability

3 Conditional Probabilities are Probabilities

4 Independence of Events

5 Conditioning As A Problem-Solving Tool

6 Pitfalls & Paradoxes

7 Advanced Part

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 9 / 60

Chain Rule

Theorem

For any events A1, . . . ,An with positive probabilities,

P(A1, . . . ,An) = P(A1)P(A2|A1)P(A3|A1,A2) · · ·P(An|A1, . . . ,An−1).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 10 / 60

Page 6: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Bayes’ Rule

Theorem

For any events A and B with positive probabilities,

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

P(B).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 11 / 60

Odds

Definition

The odds of an event A are

odds(A) =P(A)

P(Ac).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 12 / 60

Page 7: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Odds From Bayes’ Rule

Theorem

For any events A and B with positive probabilities, the odds of A afterconditioning on B are

P(A|B)

P(Ac |B)=

P(B|A)

P(B|Ac)· P(A)

P(Ac).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 13 / 60

The Law of Total Probability (LOTP)

Theorem

Let A1, . . . ,An be a partition of the sample space S (i.e., the Ai aredisjoint events and their union is S), with P(Ai ) > 0 for all i . Then

P(B) =n∑

i=1

P(B|Ai )P(Ai ).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 14 / 60

Page 8: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Inference & Bayes’ Rule

Theorem

Let A1, . . . ,An be a partition of the sample space S (i.e., the Ai aredisjoint events and their union is S), with P(Ai ) > 0 for all i . Then forany event B such that P(B) > 0, we have

P(Ai |B) =P(Ai )P(B|Ai )

P(A1)P(B|A1) + · · ·+ P(An)P(B|An).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 15 / 60

Example: Random Coin

You have one fair coin, and one biased coin which lands Heads withprobability 3/4. You pick one of the coins at random and flip it threetimes. It lands Heads all three times. Given this information, what is theprobability that the coin you picked is the fair one?

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 16 / 60

Page 9: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Outline

1 Definition & Intuition

2 Bayes’ Rule & The Law of Total Probability

3 Conditional Probabilities are Probabilities

4 Independence of Events

5 Conditioning As A Problem-Solving Tool

6 Pitfalls & Paradoxes

7 Advanced Part

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 17 / 60

Conditional Probability

Condition on an event E , we update our beliefs to be consistent withthis knowledge.

P(·|E ) is also a probability function with sample space S :I 0 ≤ P(·|E ) ≤ 1 with P(S |E ) = 1 and P(∅|E ) = 0.I If events A1,A2, . . . are disjoint, then P(∪∞j=1Aj |E ) =

∑∞j=1 P(Aj |E ).

I P(Ac |E ) = 1− P(A|E ).I Inclusion-exclusion: P(A ∪ B|E ) = P(A|E ) + P(B|E )− P(A ∩ B|E ).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 18 / 60

Page 10: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Bayes’ Rule with Extra Conditioning

Theorem

Provided that P(A ∩ E ) > 0 and P(B ∩ E ) > 0, we have

P(A|B,E ) =P(B|A,E )P(A|E )

P(B|E ).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 19 / 60

LOTP with Extra Conditioning

Theorem

Let A1, . . . ,An be a partition of the sample space S (i.e., the Ai aredisjoint events and their union is S), with P(Ai ∩ E ) > 0 for all i . Then

P(B|E ) =n∑

i=1

P(B|Ai ,E )P(Ai |E ).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 20 / 60

Page 11: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Example: Random Coin

You have one fair coin, and one biased coin which lands Heads withprobability 3/4. You pick one of the coins at random and flip it threetimes. It lands Heads all three times. If we toss the coin a fourth time,what is the probability that it will land Heads once more?

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 21 / 60

Approaches for Finding P(A|B ,C )

Think of B,C as the single event B ∩ C .

Use Bayes’ rule with extra conditioning on C .

Use Bayes’ rule with extra conditioning on B.

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 22 / 60

Page 12: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Outline

1 Definition & Intuition

2 Bayes’ Rule & The Law of Total Probability

3 Conditional Probabilities are Probabilities

4 Independence of Events

5 Conditioning As A Problem-Solving Tool

6 Pitfalls & Paradoxes

7 Advanced Part

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 23 / 60

Independence of Two Events

Definition

Events A and B are independent if

P(A ∩ B) = P(A)P(B).

If P(A) > 0 and P(B) > 0, then this is equivalent to

P(A|B) = P(A)

P(B|A) = P(B)

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 24 / 60

Page 13: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Independence vs. Disjointness

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 25 / 60

Independence of Complementary Set

Theorem

If A and B are independent, then A and Bc are independent, Ac and B areindependent, and Ac and Bc are independent.

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 26 / 60

Page 14: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Independence of Three Events

Definition

Events A, B and C are independent if all of the following equations hold:

P(A ∩ B) = P(A)P(B)

P(A ∩ C ) = P(A)P(C )

P(B ∩ C ) = P(B)P(C )

P(A ∩ B ∩ C ) = P(A)P(B)P(C ).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 27 / 60

Pairwise Independence and Independence

Consider two fair, independent coin tosses.

A: the event that the first is Heads.

B: the event that the second is Heads.

C: the event that both tosses have the same result.

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 28 / 60

Page 15: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Conditional Independence

Definition

Events A and B are said to be conditionally independent given E if:

P(A ∩ B|E ) = P(A|E )P(B|E ).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 29 / 60

Example: Conditional Independence 6=⇒ Independence

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 30 / 60

Page 16: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Example: Independence 6=⇒ Conditional Independence

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 31 / 60

Example: Conditional Independence Given E vs. E c

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 32 / 60

Page 17: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Outline

1 Definition & Intuition

2 Bayes’ Rule & The Law of Total Probability

3 Conditional Probabilities are Probabilities

4 Independence of Events

5 Conditioning As A Problem-Solving Tool

6 Pitfalls & Paradoxes

7 Advanced Part

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 33 / 60

Example: Monty Hall Problem

On the game show Let’s Make a Deal, hosted by Monty Hall, a contestantchooses one of three closed doors, two of which have a goat behind themand one of which has a car. Monty, who knows where the car is, thenopens one of the two remaining doors. The door he opens always has agoat behind it (he never reveals the car!). If he has a choice, then he picksa door at random with equal probabilities. Monty then offers thecontestant the option of switching to the other unopened door. If thecontestant’s goal is to get the car, should she switch doors?

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 34 / 60

Page 18: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Solution of Monty Hall Problem

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 35 / 60

Tree Diagram of Monty Hall Problem

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 36 / 60

Page 19: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Tree Diagram of Monty Hall Problem

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 37 / 60

Extension of Monty Hall Problem

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 38 / 60

Page 20: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Extension of Monty Hall Problem

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 39 / 60

Example: Laplace’s Rule of Succession

There are k + 1 coins in a box. When flipped, the i th coin will turn upheads with probability i/k , i = 0, 1, . . . , k. A coin is randomly selectedfrom the box and is then repeatedly flipped. If the first n flips all result inheads, what is the conditional probability that the (n + 1)st flip will dolikewise?

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 40 / 60

Page 21: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Solution of Laplace’s Rule of Succession

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 41 / 60

Example: Branching Process

A single amoeba, Bobo, lives in a pond. After one minute Bobo will eitherdie, split into two amoebas, or stay the same, with equal probability, andin subsequent minutes all living amoebas will behave the same way,independently. What is the probability that the amoeba population willeventually die out?

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 42 / 60

Page 22: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

First-Step Analysis: Branching Process

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 43 / 60

Example: Gambler’s Ruin

Two gamblers, A and B, make a sequence of dollar 1 bets. In each bet,gambler A has probability p of winning, and gambler B has probabilityq = 1− p of winning. Gambler A starts with i dollars and gambler B startswith N − i dollars; the total wealth between the two remains constantsince every time A loses a dollar, the dollar goes to B, and vice versa. Thegame ends when either A or B is ruined (run out of money). What is theprobability that A wins the game (walking away with all the money)?

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 44 / 60

Page 23: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

First-Step Analysis: Gambler’s Ruin

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 45 / 60

Outline

1 Definition & Intuition

2 Bayes’ Rule & The Law of Total Probability

3 Conditional Probabilities are Probabilities

4 Independence of Events

5 Conditioning As A Problem-Solving Tool

6 Pitfalls & Paradoxes

7 Advanced Part

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 46 / 60

Page 24: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Prosecutor’s Fallacy

In 1998, Sally Clark was tried for murder after two of her sons died shortlyafter birth. During the trial, an expert witness for the prosecution testifiedthat the probability of a newborn dying of sudden infant death syndrome(SIDS) was 1/8500, so the probability of two deaths due to SIDS in onefamily was (1/8500)2, or about one in 73 million. Therefore, he continued,the probability of Clark’s innocence was one in 73 million. Then sadly,clark was convicted of murder and sent to prison.

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 47 / 60

Defense Attorney’s Fallacy

A woman has been murdered, and her husband is put on trial for thiscrime. Evidence comes to light that the defendant had a history of abusinghis wife. The defense attorney argues that the evidence of abuse should beexcluded on grounds of irrelevance, since only 1 in 10,000 men who abusetheir wives subsequently murder them. Should the judge grant the defenseattorney’s motion to bar this evidence from trial?

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 48 / 60

Page 25: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Example of Simpson’s Paradox

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 49 / 60

Math Behind Simpson’s Paradox

If

P(A|B,C ) < P(A|Bc ,C )

,P(A|B,C c) < P(A|Bc ,C c),

then is it possible that

P(A|B) > P(A|Bc)?

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 50 / 60

Page 26: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Another Example of Simpson’s Paradox

Gender discrimination in college admissions: In the 1970s, men weresignificantly more likely than women to be admitted for graduate study atthe University of California, Berkeley, leading to charges of genderdiscrimination. Yet within most individual departments, women wereadmitted at a higher rate than men. It was found that women tended toapply to the departments with more competitive admissions, while mentended to apply to less competitive departments.

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 51 / 60

Summary : Conditional Probability & Inference

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 52 / 60

Page 27: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Outline

1 Definition & Intuition

2 Bayes’ Rule & The Law of Total Probability

3 Conditional Probabilities are Probabilities

4 Independence of Events

5 Conditioning As A Problem-Solving Tool

6 Pitfalls & Paradoxes

7 Advanced Part

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 53 / 60

Definition of Infinite Often

Definition

Given a sequence of events An, “An infinitely often” or “An i.o.” , isdefined as “∩∞n=1 ∪∞i=n Ai”.

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 54 / 60

Page 28: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Limiting Events

If An, n ≥ 1 is either an increasing or decreasing sequence of events:

limn→∞

P(An) = P( limn→∞

An).

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 55 / 60

The Borel-Cantelli Lemma Revisit

If An, n ≥ 1 is a sequence of events and

∞∑n=1

P(An) <∞,

then

P(An i.o.) = P(∩∞n=1 ∪∞i=n Ai ) = 0.

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 56 / 60

Page 29: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

The Second Borel-Cantelli Lemma

If An, n ≥ 1 is a sequence of independent events and

∞∑n=1

P(An) =∞,

then

P(An i.o.) = P(∩∞n=1 ∪∞i=n Ai ) = 1.

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 57 / 60

A Zero-One Law

If An, n ≥ 1 is a sequence of independent events, then

P(An i.o.) =

{0 if

∑∞n=1 P(An) <∞

1 if∑∞

n=1 P(An) =∞

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 58 / 60

Page 30: Lecture 2: Conditional Probability · Outline 1 De nition & Intuition 2 Bayes’ Rule & The Law of Total Probability 3 Conditional Probabilities are Probabilities 4 Independence of

Example: Coin Tossing

Toss a fair coin repeatedly (independent tosses) and let

An = {the nth toss yields a head}, n ≥ 1.

Then P(An i.o.) = 1.

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 59 / 60

Reading Assignment

Next lecture we will review discrete random variable. Please readcorresponding chapters in required textbooks:

Chapter 3 of BH

Chapter 2 of BT

Ziyu Shao (ShanghaiTech) Lecture 2: Conditional Probability September 26, 2016 60 / 60