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History of Probability Theory

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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

BUS304 – Probability Theory

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

- Risk Management and Investment

BUS304 – Probability Theory

Experiment

Experimental Outcomes

Toss a coin

Head, tail

Inspect a part

Defective, nondefective

Play a football game

Win, lose, tie

Roll a die

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.

BUS304 – Probability Theory

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}

BUS304 – Probability Theory

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= .

- E.g. E = A die shows number higher than 3

BUS304 – Probability Theory

Rules of Assigning Probabilities

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

BUS304 – Probability Theory

Basic Rules to assign probability (1)

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

BUS304 – Probability Theory

Basic Rules to assign probability (2)

- Relative Frequency of Occurrence

Number of times E occurs

N

Probability of Future Event = Relative Freq. of Past =

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 player’s percentage of made free throws. Why do you think Yao Ming has a better chance to win the free throw competition than Shaq O’Neal?
- The probability that a TV is sent back for repair? Based on past experience.
- The most commonly used in the business world.

BUS304 – Probability Theory

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.

BUS304 – Probability Theory

Exercise 4.1 (Page 137)

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”?

BUS304 – Probability Theory

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.

BUS304 – Probability Theory

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!

- Based on experience, combining different signals to make inference. No standard approach to have people agree on each other.

BUS304 – Probability Theory

Rules for complement events

- what is the a complement event?
- The Rule:

E

E

If Obama’s chance of winning the presidential campaign is assigned to be 60%, that means McCain’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

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(E1or E2) = P(E1) + P(E2) - P(E1and E2)

E1

E2

P(E1and E2) ≤P(E1)

P(E1and E2)≤P(E2)

P(E1and E2)

E1 or E2

E1

E2

P(E1or E2) ≥P(E1)

P(E1or E2)≥P(E2)

BUS304 – Probability Theory

Exercise

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

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

- If one happens, then the other cannot happen.

E2

E1

BUS304 – Probability Theory

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

BUS304 – Probability Theory

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?

- What if I know E2 happens, would you

E1

E2

BUS304 – Probability Theory

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”)=?

Bayes’ TheoremThomas Bayes

(1702-1761)

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?

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”?

BUS304 – Probability Theory

Probability Distribution x = 0, 1, or 2

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

- Examples

- 1. Roll a die twice: Let x be the number of times 4 comes up.

- 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: Let’s 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?

BUS304 – Probability Theory

Discrete vs. Continuous Random variables

Random Variables

Discrete

Continuous

- Examples:

- Examples:

BUS304 – Probability Theory

.25

0 1 2 x

Discrete Probability DistributionTwo ways to represent discrete probability distributions

Table

All the possible values of x

Probability

Graph

BUS304 – Probability Theory

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

- Step 1.1: Write down the sample space

- Draw a pair of dice, x is the random variable representing the sum of the total points.

BUS304 – Probability Theory

-2 .25

0 .50

2 .25

Measures of Discrete Random VariablesExample:

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.

BUS304 – Probability Theory

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

E(x) =-0.5+0+0.5=0

BUS304 – Probability Theory

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?

BUS304 – Probability Theory

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(x+y) = 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
- x+y is your total gain from playing the two games.

- Example: “Head -$2”, “Tail +1”

Write down the probability distribution of x+y and calculate the expected value for x+y

E(y)= -0.5

E(x)= -0.5

Is this game a fair game?

BUS304 – Probability Theory

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): 50x0.2+60x0.3+70x0.4+80x0.1=64

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.

BUS304 – Probability Theory

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

Variance and Standard deviationBUS304 – Probability Theory

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