ERG 2040 tutorial 1

1
ERG 2040 tutorial 1

• Zhu Lei

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Components of a Probability Model

• The probability theory is designed for random
  experiments. We want to know how likely a certain
  outcome would appear.
• The Sample Space the totality of the possible
  outcomes of a random experiment. (S)
• An event a collection of certain sample points,
  or a subset of the sample space. (E)

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Components of a Probability Model

• Example 1
• Prof. Liew and Prof. Yum do paper-scissors-rock
  (?????) to decide who will grade the homework.
  The rule is if some one loses, he will do all
  the works if it is a draw, they do it together.
• First, we pay attention to Prof. Liews choice
• The Sample Space is
• The Event set any combinations of these three
  choices
• Event 1 Prof. Liew chose Scissors
• E1Scissors
• Event 2 Prof. Liew did not choose rock
• E2Paper, Scissors.

Paper, Scissors, Rock
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Components of a Probability Model

• Example 1
• Prof. Liew and Prof. Yum do paper-scissors-rock
  (?????) to decide who will grade the homework.
  The rule is if some one loses, he will do all
  the works if it is a draw, they do it together.
• Second, we pay attention to Prof. Liew and Prof.
  Yum choices.
• The Sample Space is
• It contains 9 possible choices.
• The Event set any combinations of these nine
  choices
• Event 1 Prof. Liew chose Scissors while Prof.
  Yum dose not chose Paper
• E1 (s,s), (s,r)

(p,p), (p,s), (p,r), (s,p), (s,s), (s,r),
(r,p), (r,s), (r,r)
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Components of a Probability Model

• Example 1
• Prof. Liew and Prof. Yum do paper-scissors-rock
  (?????) to decide who will grade the homework.
  The rule is if some one loses, he will do all
  the works if it is a draw, they do it together.
• Next, we pay attention to the outcome of the
  game. That is who will grade the homework.
• The Sample Space is ?
• A. Prof. Yum, Prof. Liew
• B. Prof. Yum, Prof. Liew, Prof Yum
  Prof. Liew
• C. Prof. Yum grade homework while Prof.
  Liew does not,
• Prof. Liew grade homework while Prof.
  Yum does not,
• Both Prof. Yum and Prof Liew grade
  the homework

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Exclusive Vs. Independent

• Exclusive events are those from the same trial
  of the same experiment. They cannot happen
  together.
• Independent events are those from different
  experiments or different trials of one
  experiment. The outcome of one event does not
  influence the others.
• Example
• E1Prof. Liew choose paper in the first round
• E2Prof. Liew choose rock in the first round
• E3Prof. Yum choose paper in the first round
• E4Prof. Liew choose rock in the second round
• E1 and E2 are mutually exclusive, because Prof.
  Liew cannot choose both paper and stone at the
  same time, although he has two hands.

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Mutually Independent Vs. Pairwise Independent

• Example
• Let Y denote Prof. Yums choice.
• Let L denote Prof. Liews choice.
• Let C denote the outcome of the game
• Suppose Prof. Yum and Prof. Liew made their
  choice randomly.
• Is Y independent with C? P(YnC)P(Y)P(C)?
• Yes
• Are they mutually independent? Or
  P(YnLnC)P(Y)P(L)P(C)?
• No

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Independent Events

• Example
• Prof. Liew and Prof. Yum are both very smart
  guys. After several months, Prof Liew found out
  that Prof. Yum likes to use rock the most. So
  he decide to use paper to defeat Prof. Yum in
  the next round. Now is L independent with Y?
  (Yes)
• Prof. Yum found out that Prof. Liew would blink
  his eyes whenever he want to use scissors. After
  finding out this, Prof. Yum decide to use rock if
  Prof. Liew blink his eyes and use paper
  otherwise. Now is L independent with Y? (No)
• Following Prof. Yums strategy, does he still
  have chance to lose?
• Yes, he still have chance to lose

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Conditional Probability

• The conditional probability of A given B is
• Example
• 100 students took erg2040. 15 of them got Grade
  A for both homework and exams 30 of them got
  Grade A for homework 20 of them got Grade A in
  exams. For a student who did homework very well,
  what is the probability he got an A in exams?
• P(H)0.3, P(E)0.2, P(HnE)0.15
• P(E H) P(HnE)/ P(H)0.5
• This shows doing homework is very helpful!!

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Bayes rule

• The posteriori probability of Bj given A is
• Example
• It is a peaceful night. You are sleeping.
  
• One of a sudden, the fire alarm rings
• You remember the fire alarm is 99 accurate,
• or P( Alarm Fire )99..............
• You got desperate..
• At this moment, you remember a very important
  thing that may save your life

Bayes Rule!!
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Bayes rule

• Example
• The Fire alarm is 99 accurate, or P( Alarm Fire
  )99. But it can be triggered by something else,
  smoking, candle, etc, So if there is no fire, it
  can still ring with P( Alarm no fire)2. For a
  random night, the probability that the dormitory
  is on fire is very small P( Fire )0.05. Given
  the fire alarm is ringing, what is the
  probability that there is a fire?

Then you can go back to sleep
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Bayes rule
Fire
Candle
Smoking
There are many different things that can trigger
the alarm. By Bayes rule, we can infer their
possibility.
Alarm
Bomb
By the Bayes rule
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Bernoulli trials

• For n trials, the probability of exactly k
  successes and (n-k) failures
• The outcomes of different trials are independent.
• win with probability p and lose with probability
  1-p for any trial.
• The order of outcomes does not matter.
• win, win, lose, win, lose, win and lose,
  win, win are considered as the same event with
  k2.

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Bernoulli trials

• Example
• We gamble by rolling five dices. If the outcome
  contains exactly one 6, you win 1 dollar from
  me if not, I win 1 dollar from you. Will you
  play this with me?
• p1/6, the outcome of one dice is 6.
• q1-p5/6, the outcome of one dice is 1, 2,
  3, 4, 5.
• Pr you win pexactly one dice is 6

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Bernoulli trials

• Example
• We gamble by rolling five dices. If the outcome
  contains exactly two 1s and two 2s, you win
  one dollar from me. If it contains exactly three
  2s, I win one dollar from you.
• p11/6, the outcome of one dice is 1.
• p21/6, the outcome of one dice is 2.

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A question from homework

• Three couples (husbands and their wives) must sit
  at a round table in such a way that no husband is
  placed next to his wife. How many configurations
  exist?