Probability 2.3 Independence Anna Karlin Most slides by Alex Tsun - - PowerPoint PPT Presentation

Probability

2.3 Independence

Anna Karlin Most slides by Alex Tsun

Agenda

• Chain Rule
• Independence
• Conditional Independence

not

in class

Chain Rule (Idea)

Have a Standard 52-Card Deck.

• 4 Suits (Clubs,

Diamonds, Hearts, Spades)

• 13 ranks (A, 2, 3, …,

9, 10, J, Q, K)

Chain Rule (Idea)

Have a Standard 52-Card Deck. Shuffle It, and draw the top 3 cards. What is P ( ) = P(A, B, C)? A: Ace of Spades First B: 10 of Clubs Second C: 4 of Diamonds Third

Chain Rule (Idea)

Have a Standard 52-Card Deck. Shuffle It, and draw the top 3 cards. What is P ( ) = P(A, B, C)? A: Ace of Spades First B: 10 of Clubs Second C: 4 of Diamonds Third

PIA

PCBfA

e P

c

An B

Chain Rule (Idea)

Have a Standard 52-Card Deck. Shuffle It, and draw the top 3 cards. (uniform probability space). What is P ( ) = P(A, B, C)?

A: Ace of Spades First B: 10 of Clubs Second C: 4 of Diamonds Third

A B C

w

E

AnBnc

AnB

Chain Rule

peps A

p y

c

Chain Rule (Idea)

Have a Standard 52-Card Deck. Shuffle It, and draw the top 3 cards. (uniform probability space). What is P ( ) = P(A, B, C)?

A: Ace of Spades First B: 10 of Clubs Second C: 4 of Diamonds Third

Prf E

Irl

Irl

52 51.50

TT

PB P

2nd

card

D

d LT

P

AnB

i

si

so

Fun with cards

Two people, A and B, are playing the following game. A 6-sided die is thrown and each time it’s thrown, regardless of the history, it is equally likely to show any of the six numbers If it shows 5, A wins. If it shows 1, 2 or 6, B wins. Otherwise, they play a second round and so on. What is Pr(A wins on 4th round)?

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stIsIs

PCBlAI

PrfAy

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so IT

will

be

• n

section

worksheet

use

chain rule

The need for independence

P

HAB

The need for independence

Independence

Independence

• Toss a coin 3 times. Each of 8 outcomes equally likely.

Define

• A = {at most one T} = {HHH, HHT, HTH, THH}
• B = {at most 2 Heads}= {HHH}c
• Are A and B independent?

AAB

HHT HTH THH PrlA

PrlB

P4AnB

a

8t

b

Is

6

PFA PCB

PlAnB

42 37

not indep

d

F

Network Communication

A B C D

p r q s

Each link works with the probability given, independently. What’s the probability A and D can communicate?

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PCC tPCD

T

toppathworks

Plt

PplBBjBpD p

Be bottom

palm

works

whence ED

p9

are independent

PCB

ros

PCCND

P

PCD

PFFworking

path

P TUB

P T tPlB

PfthB

sa

t

I Pri if Y.SI

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Network Communication

A B C D

p r q s

Each link works with the probability given, independently. What’s the probability A and D can communicate?

P D PCD14

Using independence to define a probabilistic model

• We can define our probability model via independence.
• Example: suppose a biased coin comes up heads with

probability 2/3, independent of other flips.

• Sample space: sequences of 3 coin tosses.
• Pr (HHH)=?
• Pr (TTT) = ?
• Pr (HHT) = ?
• Pr (HTH) = ?
• Pr (2 heads) = ?

e

r

Hit

Iii

Iii

4

50

Prc

I

d E5

Probability

3.1 Discrete Random Variables Basics

Anna Karlin Most slides by Alex Tsun

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Agenda

• Intro to Discrete Random Variables
• Probability Mass Functions
• Cumulative Distribution function

Flipping two coins

Fandom variable

Flipping two coins

r

setypossible

values X

si

Ft

1 191,23

Random Variable

Random Variable