Probability - University of Washington · 2020. 10. 12. · Chain Rule (Idea) Have a Standard...
Transcript of Probability - University of Washington · 2020. 10. 12. · Chain Rule (Idea) Have a Standard...
-
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