1. 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
BUS304 – Probability Theory

2. Applications of Probability Theory
The World is full of uncertainty!
Knowing probability theory is important !
Gambling:
Poker games, lotteries, etc.
Weather report:
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
Reading information from the chart.
BUS304 – Probability Theory

3. Concept: Experiment and event
Experiment: A process of obtaining well-defined outcomes for uncertain events
Event: A certain outcome in an experiment
Example:
Roll a die
Win, lose, tie
Play a football game
Defective, nondefective
Inspect a part
Head, tail
Toss a coin
Experimental Outcomes
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.
BUS304 – Probability Theory

4. Basic Rules to assign probability (1)
Classical probability Assessment:
Basic Rules to assign probability (1)
where:
E refers to a certain event.
P(E) represent the probability of the event E
Exercise:
Decide the probability of the following events
Get a card higher than 10 from a bridge deck
Get a sum higher than 11 from throwing a pair of dice.
John and Mike both randomly pick a number from 1-5, what is the chance that these two numbers are the same?
P(E) =
Number of ways E can occur
Total number of ways
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
BUS304 – Probability Theory

5. Basic Rules to assign probability (2)
Find the relative frequency => probability
Relative Frequency of Occurrence
Relative Freq. of Ei =
Number of times E occurs N
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.
Reading information from the chart.
BUS304 – Probability Theory

6. Exercise
A clerk recorded the number of patients waiting for service at 9:00am on 20 successive days
Assign the probability that there are at most 2 agents waiting at 9:00am.
Number of waiting
Number of Days Outcome Occurs 2 1 5 6 3 4
Total
BUS304 – Probability Theory

7. 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.
6.31 days
Always try to use as much information as possible.
As the world is changing dramatically, people are more and more rely upon subjective assessment.
BUS304 – Probability Theory

8. Rules for complement events
what is the a complement event?
The Rule: E 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?
BUS304 – Probability Theory

9. More Exercise (homework)
Page 137
Problem 4.2 (a) (b) (c)
Problem 4.5
Problem 4.8 (a)
Problem 4.10
BUS304 – Probability Theory

10. BUS304 – Probability Theory
Composite Events
E = E1 and E2
=(E1 is observed) AND (E2 is also observed)
E = E1 or E2
= Either (E1 is observed) Or (E2 is observed)
More specifically, P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2) E1 E2
P(E1 and E2) ≤ P(E1)
P(E1 and E2) ≤ P(E2)
P(E1 and E2)
E1 or E2 E1 E2
P(E1 or E2) ≥ P(E1)
P(E1 or E2) ≥ P(E2)
BUS304 – Probability Theory

11. BUS304 – Probability Theory
Exercise
Male
Female
Total
Under 20
168
208
376
20 to 40
340
290
630
Over 40
170
160
330
678
658
1336
What is the probability of selecting a person who is a male?
What is the probability of selecting a person who is under 20?
What is the probability of selecting a person who is a male and also under 20?
What is the probability of selecting a person who is either a male or under 20?
BUS304 – Probability Theory

12. 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
mutually exclusive.
Complement Events E1 E2
BUS304 – Probability Theory

13. Rules for mutually exclusive events
If E1 and E2 are mutually exclusive, then
P(E1 and E2) = ?
P(E1 or 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) E1 E2
BUS304 – Probability Theory

14. 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
change your estimation? E1 E2
BUS304 – Probability Theory

15. BUS304 – Probability Theory
Bayes’ Theorem
Conditional Probability Rule:
Example:
P(“Male”)=? P(“GPA 3.0”)=?
P(“Male” and “GPA<3.0”)=? P(“Female” and “GPA 3.0”)=?
P(“GPA<3.0” | “Male”) = ? P (“Female” | “GPA 3.0”)=?
Thomas Bayes
( )
GPA3.0
GPA<3.0
Male
282
323
Female
305
318
BUS304 – Probability Theory

16. BUS304 – Probability Theory
Independent Events If
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.
How to prove?
BUS304 – Probability Theory 16

17. BUS304 – Probability Theory
Exercise
Calculate the following probabilities:
Prob of getting 3 heads in a row?
Prob of a “double-six”?
Prob of getting a spade card which is also higher than 10?
Data shown from the following table. Decide whether the following events are independent?
“Selecting a male” versus “selecting a female”?
“Selecting a male” versus “selecting a person under 20”?
Male
Female
Under 20
168
208
20 to 40
340
290
Over 40
170
160
BUS304 – Probability Theory