prob09 lecture3 revised

7/31/2019 Prob09 Lecture3 Revised

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Lecture 3

Conditional Probability, Independence

and Sequential Experiments

Last Time

Probability Axioms

Some Consequences of the Axioms

Conditional Probability

Reading Assignment: Sections 1.1-1.5

Probability & Stochastic Processes

Yates & Goodman (2nd Edition) NTUEE SCC_03_20083- 1


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Lecture 3: Probabilities and Experiments

This Week

Independence

Sequential Experiments &Tree Diagrams

Counting Methods

Independent Trials

Reliabilit Methods

Discrete Random Variables

Definitions

Probability Mass Function





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Lecture 3:

Next Week

Discrete Random Variables

Probability Mass Function

Family of D.R.Vs

Cumulative Distribution Function

Averages





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Diaconis, P., Holmes, S., and Montgomery, R. (2007)

Dynamical Bias in the Coin Toss. SIAM Review 49, 211-235.

Persi Diaconis (Jan. 31, 1945 - , )

Sunseri Prof. of Statistics and Mathematics, Stanford U.

born into a family of professional musicians

left home at 14 to travel with magician Dai Vernon

at 16 on his own as a magician for 8 years

a friend recommended a probability book by Feller and then found that he couldn't read it

so enrolled in N.Y. City College at night and got degree in mathematics in 2.5 years

got PhD in statistics, Harvard, in 3 years Stanford

an expert at deception

"I work from seven A.M. to midnight each day. I'm always doing mathematics."


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0

sender receiver

0p

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11

q


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Conditional Probability (Graphic Interpretation)

S

A B1



B2

P(A|B1) = ?

P(B1|A) = ?


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Semiconductor Process Control

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Example 1.23 again,Q: E1 = {1,2}, E2={2,3}, E3={1,3},

E1 & E3, E2&E3, E1 &E2 independent?How about E1&E2&E3?


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Q: 20!=

=2.43x 1910


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VLSI Testing example

A semiconductor wafer has M VLSI chips on it and these chipshave the same circuitry. Each VLSI chip consists ofN

interconnected transistors. A transistor may fail (not function

properly) with a probability p because of its fabrication process,

which we assume to be independent among individual

Semiconductor Yield Analysis

transistors. A chip is considered a failure if there are n or moretransistor failures.

Derive the probability that a chip is good.




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Semiconductor Yield Analysis: Root Cause Diagnosis

Now suppose that the value of a current Iof the chip depends ontransistor 1. If transistor 1 fails, we will observe an abnormal I

value with a probability q and a normal Ivalue with a

probability 1-q; if transistor 1 is good, we will observe an

normal Ivalue with a probability rand an abnormal Ivalue with

a probability 1-r. What is the probability that you measure anabnormal Ivalue? When the Ivalue you measured is normal,

what is the probability that transistor 1 actually fails?




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Independent Trials

Gamblers Ruin Problem

Two gamblers A and B with a and b dollars play the game of faircoin toss with wager of $1 per game until one of them runs out

of money. What is the probability that A runs out of money?

Ans: b/(a+b), why?


Yates & Goodman (2nd Edition) NTUEE SCC_03_2008

Let p(i) be the probability that a gambler starts with i dollars and run out of money eventually.After a coin toss, the gambler either wins or loses and starts from i+1 or i-1 dollars, so

p(i )= (1/2) p(i+1) + (1/2) p(i-1) .Also, p(0) = 1, p(a+b)= 0, so

p(i+1 )- p(i ) = p(i) p(i-1)Finally, we have

p(i)= (a+b-i)/(a+b),p(a) = b/(a+b).

For an unfair coin toss game, we can also find the probability that gambler A runs out of money

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Jailers Paradox: to tell or not to tell

Example Alex, Ben, and Tim are three prisoners, one of

whom is scheduled to die. Alex asks a jailer to tell him whowill be freed so that Alex could ask him to bring a letter to

Alexs wife. If the jailer tells Alex that which one of Ben and

Tim is going to be freed, will this change the probability of

Alex dying?



Let A, B, T, and J be the event that Alex, Ben, and Tim willdie and the event that a jailer told Alex that Tim will be freed.

Then

P(A|J) = P(J|A)P(A)/[P(J|A)P(A) + P(J|B)P(B) + P(J|T)P(T)]

= (1/2)(1/3)/[(1/2)(1/3) + 1(1/3) + 0(1/3)]

= 1/3.

Q: Relation to independence?


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Code Pattern Detection Example

One bit arrives in each clock0 with prob. P 1 with Prob. 1-p

Prob.{001 is not detected in n clocks} = ?

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Chapter 2 Discrete RandomVariables


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Example 2.A1

(1) Toss a coin

(2) Gender at birth

(3) Random walk

Q: Probability space of each experiment?


prob09 lecture3 revised

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