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History of Probability Theory

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History of Probability Theory

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- 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

- 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

Example:

Experiment

Experimental Outcomes

Toss a coin

Head, tail

Inspect a part

Defective, nondefective

Play a football game

Win, lose, tie

Roll a die

- 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

- 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

- 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

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

BUS304 – Probability Theory

- 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

BUS304 – Probability Theory

- 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

- 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

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

- 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

- 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

- 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

- 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

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

- 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

- 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

- 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

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

(1702-1761)

BUS304 – Probability Theory

- 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

- 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

- 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

Random Variables

Discrete

Continuous

- Examples:

- Examples:

BUS304 – Probability Theory

.50

.25

0 1 2 x

Two ways to represent discrete probability distributions

Table

All the possible values of x

Probability

Graph

BUS304 – Probability Theory

- 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

x P(x)

-2 .25

0 .50

2 .25

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.

BUS304 – Probability Theory

- 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

- 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

- 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?

- 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

- 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?

- 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?

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

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

BUS304 – Probability Theory

- Page 4.66

BUS304 – Probability Theory