Transcript Ch 4

History of Probability Theory
 Started in the year of 1654
 a well-known gambler, De Mere asked a question to Blaise
Pascal
Whether to bet on the following event?
“To throw a pair of dice 24 times, if a ‘double six’ occurs at least once, then win.”
correspond
Blaise Pascal
Pierre Fermat
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Applications of Probability Theory
 Gambling:
 Poker games, lotteries, etc.
 Weather report:
The World is full of uncertainty!
Knowing probability theory is important !
 Likelihood to rain today
 Power of Katrina
 Statistical Inferential
 Risk Management and Investment
• Value of stocks, options, corporate debt;
• Insurance, credit assessment, loan default
 Industrial application
• Estimation of the life of a bulb, the shipping date, the daily production
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Concept: Experiment and event
 Experiment: A process of obtaining well-defined outcomes for
uncertain events
Example:
Experiment
Experimental Outcomes
Toss a coin
Head, tail
Inspect a part
Defective, nondefective
Play a football game
Win, lose, tie
Roll a die
 Event: A certain outcome in an experiment
Example:
 Two heads in a row when you flip a coin three times;
 At least one “double six” when you throw a pair of dice 24 times.
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Basic Rules to assign probability (1)
Classical probability Assessment:
Number of ways E can occur
P(E) =
Total number of ways
where:
• E refers to a certain event.
• P(E) represent the probability of the
event E
Exercise:
Decide the probability of the
following events
1. Get a card higher than 10 from a
bridge deck
2. Get a sum higher than 11 from
throwing a pair of dice.
3. John and Mike both randomly pick
When to use this rule?
When the chance of each way is the same:
e.g. cards, coins, dices, use random number
generator to select a sample
a number from 1-5, what is the
chance that these two numbers
are the same?
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Basic Rules to assign probability (2)
 Relative Frequency of Occurrence
Number of times E occurs
N
Relative Freq. of Ei =
Find the relative frequency => probability
Examples:
 If a survey result says, among 1000 people, 500 of them think the new 2GB ipod
nano is much better than the 20GB ipod. Then you assign the probability that a
person like Nano better is 50%.
 A basketball player’s proportion of made free throws
 The probability that a TV is sent back for repair
 The most commonly used in the business world.
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Exercise
A clerk recorded the number of patients waiting
for service at 9:00am on 20 successive days
Number of waiting
Number of Days Outcome Occurs
0
2
1
5
2
6
3
4
4
3
Total
20
Assign the probability that there are at most 2 agents waiting at 9:00am.
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Basic Rules to assign probability (3)
 Subjective Probability Assessment
 Subjective probability assessment has to be used when there is not
enough information for past experience.
 Example1: The probability a player will make the last minute shot (a
complicated decision process, contingent on the decision by the
component team’s coach, the player’s feeling, etc.)
 Example2: Deciding the probability that you can get the job after the
interview.
•
•
•
•
•
Smile of the interviewer
Whether you answer the question smoothly
Whether you show enough interest of the position
How many people you know are competing with you
Etc.
 Always try to use as much information as possible.
 As the world is changing dramatically, people are more and more rely upon subjective assessment.
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Rules for complement events
 what is the a complement event?
E
E
 The Rule: P( E )  1  P(E)
If Bush’s chance of winning is assigned to be 60% before the election, that means
Kerry’s chance is 1-60% = 40%.
If the probability that at most two patients are waiting in the line is 0.65, what is the
complement event? And what is the probability?
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More Exercise (homework)
Page 137
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Problem 4.2 (a) (b) (c)
Problem 4.5
Problem 4.8 (a)
Problem 4.10
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Composite Events
 E = E1 and E2
=(E1 is observed) AND (E2 is also observed)
E1
E2
P(E1 and E2) ≤ P(E1)
P(E1 and E2) ≤ P(E2)
P(E1 and E2)
 E = E1 or E2
= Either (E1 is observed) Or (E2 is observed)
E1
E1 or E2
E2
P(E1 or E2) ≥ P(E1)
P(E1 or E2) ≥ P(E2)
More specifically, P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2)
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Exercise
Male
Female
Total
Under 20
168
208
376
20 to 40
340
290
630
Over 40
170
160
330
Total
678
658
1336
1. What is the probability of selecting a person who is a male?
2. What is the probability of selecting a person who is under 20?
3. What is the probability of selecting a person who is a male and
also under 20?
4. What is the probability of selecting a person who is either a male
or under 20?
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Mutually Exclusive Events
 If two events cannot happen simultaneously, then these
two events are called mutually exclusive events.
 Ways to determine whether two events are mutually
exclusive:
 If one happens, then the other cannot happen.
Examples:
 Draw a card, E1 = A Red card, E2 = A card of club
 Throwing a pair of dice, E1 = one die shows
E2 = a double six.
 All elementary events are
E2
E1
mutually exclusive.
 Complement Events
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Rules for mutually exclusive events
If E1 and E2 are mutually exclusive, then
 P(E1 and E2) = ?
 P(E1 or E2) = ?
E1
E2
Exercise:
 Throwing a pair of dice, what is the probability that I
get a sum higher than 10?
 E1: getting 11
 E2: getting 12
 E1 and E2 are mutually exclusive.
 So P(E1 or E2) = P(E1) + P(E2)
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Conditional Probabilities
 Information reveals gradually, your estimation changes
as you know more.
 Draw a card from bridge deck (52 cards). Probability of
a spade card?
 Now, I took a peek, the card is black, what is the probability of a
spade card?
 If I know the card is red, what is the probability of a spade card?
 What is the probability of E1?
 What if I know E2 happens, would you
E1
E2
change your estimation?
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Bayes’ Theorem
 Conditional Probability Rule:
P E1 and E2 
P E1 | E2  
P E2 
Example:
GPA3.0
GPA<3.0
Male
282
323
Female
305
318
P(“Male”)=?
P(“Male” and “GPA<3.0”)=?
P(“GPA<3.0” | “Male”) = ?
Thomas Bayes
(1702-1761)
P(“GPA 3.0”)=?
P(“Female” and “GPA 3.0”)=?
P (“Female” | “GPA 3.0”)=?
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Independent Events
 If PE1 | E2   PE1 
then we say that “Events E1 and E2 are independent”.
That is, the outcome of E1 is not affected by whether E2
occurs.
 Typical Example of independent Events:
 Throwing a pair of dice, “the number showed on one die” and
“the number on the other die”.
 Toss a coin many times, the outcome of each time is
independent to the other times.
Independen t : PE1 and E2   PE1  PE2 
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How to prove?
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Exercise
1. Calculate the following probabilities:
a) Prob of getting 3 heads in a row?
b) Prob of a “double-six”?
c) Prob of getting a spade card which is also higher than 10?
2. Data shown from the following table. Decide whether the following
events are independent?
a) “Selecting a male” versus “selecting a female”?
b) “Selecting a male” versus “selecting a person under 20”?
Male
Female
Under 20
168
208
20 to 40
340
290
Over 40
170
160
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