History of Probability Theory

- Started in the year of 1654
- De Mere (a well-known gambler) asked a question

to Blaise Pascal (a mathematician)

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
- Many more in modern business world
- 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

Concept Experiment and event

- Experiment A process that
- produces a single outcome
- whose result cannot be
- predicted with certainty.
- Event A certain outcome obtained in an

experiment.

- Example of an event (description of outcome)
- 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.

Description of Events

- Elementary Events
- The most rudimentary outcomes resulting from a

simple experiment - Throwing one die, obtaining a is an

elementary event - Denoted as e1, e2, , en
- Note the elementary events cannot be further

divided into smaller events. - e.g. flip a coin twice, how many elementary

events you expect to observe? - getting one head one tail is NOT an elementary

event. - Elementary events are HH, HT, TH, TT

Description of Events

- Sample Space
- Collection of all elementary outcomes
- In many experiments, identifying sample space is

important. - Write down the sample space of the following

experiments - throwing a pair of dice.
- flipping a coin three times.
- drawing two cards from a bridge deck.
- An event (denoted as E), can be represented as a

combination of elementary events. - E.g. E A die shows number higher than 3
- Elementary events e1 e2 e3

.

Rules of Assigning Probabilities

- Three rules are commonly used
- Classical Probability Assessment
- Relative Frequency Assessment
- Subjective Probability Assessment

Basic Rules to assign probability (1)

- Classical probability Assessment

- 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?

Number of Elementary Events Total number of

Elementary Events

P(E)

- where
- E refers to a certain event.
- P(E) represents the probability of the event E

When to use this rule? When the chance of each

elementary event is the same e.g. cards, coins,

dices, use random number generator to select a

sample

Basic Rules to assign probability (2)

- Relative Frequency of Occurrence

- Examples
- If a survey result says, among 1000 people, 600

prefer iphone to ipod touch, then you assign the

probability that the next person you meet will

like iphone is 60. - A basketball players percentage of made free

throws. Why do you think Yao Ming has a better

chance to win the free throw competition than

Shaq ONeal? - The probability that a TV is sent back for

repair? Based on past experience. - The most commonly used in the business world.

Exercise

- A clerk recorded the number of patients waiting

for service at 900am on 20 successive days

Assign the probability that there are at most 2

agents waiting at 900am.

Number of waiting Number of Days Outcome Occurs

0 2

1 5

2 6

3 4

4 3

Total 20

Exercise 4.1 (Page 137)

Male Female

Under 20 168 208

20 to 40 340 290

Over 40 170 160

Elementary Events? Sample Space? a) Probability

that a customer is a male? b) Probability that

a customer is 20 to 40 years old? c)

Probability that a customer being 20 to 40 years

old and a male?

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

teams coach, the players 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.

Summary of Basic Approaches

- Classical Rule
- Elementary events have equal odds
- Relative Frequency
- Use relative frequency table. Probability

assigned based on percentage of occurrence. - Subjective
- Based on experience, combining different signals

to make inference. No standard approach to have

people agree on each other. - No matter what method used, probability cannot

be higher than 1 or lower than 0!

Rules for complement events

- what is the a complement event?
- The Rule

E

E

If Obamas chance of winning the presidential

campaign is assigned to be 60, that means

McCains 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?

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

E2

P(E1 or E2) P(E1) P(E1 or E2) P(E2)

Exercise

Male Female Total

Under 20 168 208 376

20 to 40 340 290 630

Over 40 170 160 330

Total 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?

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

E2

E1

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)

E2

E1

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 GPAlt3.0)? P(Female and GPA

?3.0)? - P(GPAlt3.0 Male) ? P (Female GPA

?3.0)?

Thomas Bayes (1702-1761)

GPA?3.0 GPAlt3.0

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.

How to prove?

20

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

Probability Distribution

- Random Variable
- A variable with random (unknown) value.

Examples

1. Roll a die twice Let x be the number of

times 4 comes up. x 0, 1, or 2

2. Toss a coin 5 times Let x be the number of

heads x 0, 1, 2, 3, 4, or 5

3. Same as experiment 2 Lets say you pay your

friend 1 every time head shows up, and he pays

you 1 otherwise. Let x be amount of money you

gain from the game. What are the possible

values of x?

Discrete vs. Continuous Random variables

Random Variables

Continuous

Discrete

Examples

Examples

Number of students showed up next time

The temperature tomorrow

Number of late apt. rental payments in Oct.

The total rental payment collected by Sep 30

Your score in this coming mid-term exam

The expected lifetime of a new light bulb

Discrete Probability Distribution

X P(X)

0 0.25

1 0.5

2 0.25

Table

All the possible values of x

- Two ways to represent discrete probability

distributions

Probability

Graph

Exercise

- Describe the probability distribution of the

random variables - Draw a pair of dice, x is the random variable

representing the sum of the total points. - Step 1 Write down all the possible values in

left column - Step 1.1 Write down the sample space
- Step 2 Write down the corresponding

probabilities

Measures of Discrete Random Variables

Example What is your expected gain when you

play the flip-coin game twice?

- Expected value of a discrete distribution
- An weighted average, taking into account the

probability - The expected value of random variable x is

denoted as E(x)

E(x) ?xi P(xi) E(x) x1P(x1) x2P(x2)

xnP(xn)

E(x) (-2) 0.25 0 0.5 2 0.25

0

Your expected gain is 0! a fair game.

Spreadsheet to compute the expected value

- Step1 develop the distribution table according

to the description of the problem. - Step2 add one (3rd) column to compute the

product of the value and the probability - Step3 compute the sum of the 3rd column ? The

Expected Value

x P(x) xP(x)

-2 0.25 -2.25-0.5

0 0.5 00.50

2 0.25 20.250.5

E(x) -0.500.50

Exercise

- You are working part time in a restaurant. The

amount of tip you get each time varies. Your

estimation of the probability is as follows - You bargain with the boss saying you want a more

fixed income. He said he can give you 62 per

night, instead of letting you keep the tips.

Would you want to accept this offer?

per night Probability

50 0.2

60 0.3

70 0.4

80 0.1

More Exercise

- Buy lottery price 10
- With 0.0000001 chance, you can win 1million
- With 0.001 chance, you can win 1000
- With 0.1 chance, you can win 50
- What is the expected gain of buying this lottery

ticket? - Is buying lottery a fair game?

Rule for expected value

- If there are two random variables, x and y. Then
- E(xy) E(x) E(y)
- Example Head -2, Tail 1
- x is your gain from playing the game the first

time - y is your gain from playing the game the second

time - xy is your total gain from playing the two games.

Write down the probability distribution of xy

and calculate the expected value for xy

x P(x)

-2 0.5

1 0.5

y P(y)

-2 0.5

1 0.5

E(y) -0.5

E(x) -0.5

Is this game a fair game?

Exercise

- Assume that the expected payoff of playing the

slot machine is -0.04 cents - What is the expected payoff when playing 100

times? 10,000 times?

Measure of risk-- variance

- Two games
- Flip a coin once, if head then you get 1,

otherwise you pay 1 - Flip a coin once, if head then you get 100,

otherwise you pay 100 - Which game will you choose?
- Three basic types of people
- Risk-lover
- Risk-neutral
- Risk-averse

What is your type?

Measures variance

Step 1 develop the probability distribution

table. Step 2 compute the mean E(x)

50x0.260x0.370x0.480x0.164 Step 3 compute

the distance from the mean for each value

(x-E(x)) Step 4 square each distance (x

E(x))2 Step 5 weight the squared distance

(x-E(x))2P(x) Step 6 sum up all the weighted

square distance ? variance

- Variance a weighted average of the squared

deviation from the expected value.

x P(x) x E(x) (x-E(x))2 (x-E(x))2P(x)

50 0.2 50-64-14 (-14)2196 1960.239.6

60 0.3 -4 16 4.8

70 0.4 6 36 14.4

80 0.1 16 256 25.6

84.4 (sum of above)

Variance and Standard deviation

- Variance
- The variance of a random variable has the same

meaning as the variance of population - Calculation is the same as calculating population

variance using a relative frequency table. - Written as var(x) or

- Standard deviation of a random variable
- Same of the population standard deviation
- Calculate the variance
- Then take the square root of the variance.
- Written as sd(x) or
- e.g. for the example on page 10

More exercise

- Page 4.66