BUS304 – Probability Theory1 History of Probability Theory Started in the year of 1654 a well-known gambler, De Mere asked a question to Blaise Pascal

BUS304 – Probability Theory 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.”

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


Applications of Probability Theory 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

The World is full of uncertainty!Knowing probability theory is important !


Concept: Experiment and event Experiment: A process of obtaining well-defined outcomes for

uncertain events

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.

Example:

Roll a die

Win, lose, tiePlay a football game

Defective, nondefectiveInspect a part

Head, tailToss a coin

Experimental OutcomesExperiment


Basic Rules to assign probability (1)

P(E) =Number of ways E can occur

Total number of ways

Classical probability Assessment:

where:• E refers to a certain event. • P(E) represent the probability of the event E

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

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

a number from 1-5, what is the

chance that these two numbers

are the same?



Relative Frequency of Occurrence

Relative Freq. of Ei =Number of times E occurs

N

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.


ExerciseA 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 20Assign the probability that there are at most 2 agents waiting at

9:00am.



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.


Rules for complement events

what is the a complement event?

The Rule:

EE

P(E)1)EP(

If Bush’s chance of winning is assigned to be 60% before the election, that means Kerry’s chance is 1-60% = 40%. 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? 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?


More Exercise (homework)Page 137

Problem 4.2 (a) (b) (c) Problem 4.5 Problem 4.8 (a) Problem 4.10

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 and E2) ≤ P(E1)

P(E1 and E2) ≤ P(E2)

E1 E2E1 or E2P(E1 or E2) ≥ P(E1)

P(E1 or E2) ≥ P(E2)

Exercise


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?

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?

Male Female Total

Under 20 168 208 376

20 to 40 340 290 630

Over 40 170 160 330

Total 678 658 1336


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

E2E1


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)

E2E1

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

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”)=?


2

2121

and |

EP

EEPEEP

Thomas Bayes (1702-1761)

Thomas Bayes (1702-1761)GPA3.0 GPA<3.0

Male 282 323

Female 305 318

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.

121 | EPEEP

2121 and :tIndependen EPEPEEP

16

How to prove? How to prove?

16BUS304 – Probability Theory

Exercise

Male Female

Under 20 168 208

20 to 40 340 290

Over 40 170 160


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”?

Documents

BUS304 – Probability Theory1 History of Probability Theory Started in the year of 1654 a well-known gambler, De Mere asked a question to Blaise Pascal