Probability2.3 Independence

Anna KarlinMost 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 FirstB: 10 of Clubs SecondC: 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 FirstB: 10 of Clubs SecondC: 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 FirstB: 10 of Clubs SecondC: 4 of Diamonds Third

A B Cw E AnBncAnB

Chain Rule

peps Ap 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 FirstB: 10 of Clubs SecondC: 4 of Diamonds Third

Prf E Irl

Irl 52 51.50

TTPB P2ndcard

Dd LT

P AnB

i si so

Fun with cardsTwo 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 numbersIf 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)?

rgp fmbjml.it yEostIsIsPCBlAI PrfAy I so IT

will be on 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?

14

AAB HHT HTH THH PrlAPrlB P4AnB

a 8tb Is 6

PFA PCB PlAnB 42 37not indep

d F

Network Communication AB

CD

p

r

q

sEach link works with the probability given, independently. What’s the probability A and D can communicate? PCUD PCC tPCDT toppathworks Plt PplBBjBpD pBe bottompalmworks whence ED

p9 are independentPCB ros

PCCND P PCDPFFworkingpath P TUB P T tPlB PfthB sa

t

I Pri if Y.SI pEspus7 aP97ltrs nnodffndep0

Network Communication AB

CD

p

r

q

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

P D PCD14 0

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) = ?

17

e

r Hit

Iii

Iii450Prc I

d E5

Probability3.1 Discrete Random Variables Basics

Anna KarlinMost slides by Alex Tsun

P HAD PLATH PITHH T3 1353435 h's PRCH.TL

PCEDwEePriw3f5t3IPgI.PLIg IzIFI's

Agenda● Intro to Discrete Random Variables● Probability Mass Functions● Cumulative Distribution function

Flipping two coins

Fandom variable

Flipping two coins

rsetypossiblevalues X

siFt 1 191,23

Random Variable

Random Variable