Chapter 4. Probability: The Study of Randomness 1.

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Chapter 4. Probability: The Study of Randomness

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Uses of Probability• Gambling• Business

– Product preferences of consumers– Rate of returns on investments

• Engineering– Defective parts

• Physical Sciences– Locations of electrons in an atom

• Computer Science– Flow of traffic or communications

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4.1: Randomness - Goals• Be able to state why probability is useful.• Be able to state what randomness and probability

mean.• Be able to identify where randomness occurs in

particular situations.• Be able to state when trials are independent.

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Trial and Experiment

• Trial• Experiment

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Random and Probability

• We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

• The probability of any outcome of a chance process is the proportion of times the outcome would occur in a very long series of repetitions.

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Frequentist Interpretation

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Independent

• Independent: the outcome of each situation is not influenced by the result of the previous trial

• Examplea) What is the probability of drawing a heart?b) What is the probability that I will draw a

heart on the second draw?

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4.2: Probability Models - Goals• Be able to write down a sample space in specific

circumstances.• Be able to state and apply the five probability rules

(this goal will reappear later)• Be able to determine what type of probability is

given in a certain situation.• Be able to assign probabilities assuming an equally

likelihood assumption.

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Probability Models

• The sample space S of a chance process is the set of all possible outcomes.

• An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.

• A probability model is a description of some chance process that consists of two parts: a sample space S and a probability for each outcome.

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Sample SpaceWhat is the sample space in the following

situations? Are each of the outcomes equally likely?

a) I roll one 4-sided die.b) I roll two 4-sided dice.c) I toss a coin until the first head appears.d) A mortgage can be classified as fixed rate (F) or

variable (V) and we are considering 2 houses.e) The number of minutes that a college student

uses their cell phone in a day.

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Probability Rules

Rule 1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1.

Rule 2. If S is the sample space in a probability model, then P(S) = 1.

Rule 3. addition rule for disjoint events: If A and B are disjoint, P(A or B) = P(A) + P(B).

Rule 4: The complement of any event A is the event that A does not occur, written AC.

P(AC) = 1 – P(A).

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Examples: Probability Rules

Additivity:a) Roll two 4-sided dice: What is the probability

that the sum is 2 or 3?b) Mortgage: What is the probability that both

houses have the same type of mortgage?

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Examples: Probability Rules

Compliment:c) Roll two 4-sided dice: What is the probability

that the sum is greater than 2?d) Mortgage: What is the probability that both

houses do not have fixed mortgages?

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Types of Probabilities

• Subjective• Empirical

• Theoretical (equally likely)

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Example: Types of ProbabilitiesFor each of the following, determine the type of probability and then answer the question.1) What is the probability of rolling a 2 on a fair

4-sided die?2) What is the probability of having a girl in the

following community?

3) What is the probability that Purdue Men’s Basketball team will beat IU later this season?

Girl 0.52Boy 0.48

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Examples: Legitimate Probabilities

Which of the following probabilities are legitimate? Why or why not?

Outcome #1 #2 #3 #4 #5

1 0.25 0 0.1 0.5 1.12 0.25 0.1 0.1 -0.2 0.13 0.25 0.5 0.1 0.3 0.14 0.25 0.4 0.1 0.4 0.1

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Probability RulesRule 5. Multiplication Rule for Independent

Events. Two events A and B are independent if knowing

that one occurs does not change the probability that the other occurs.

If A and B are independent:P(A and B) = P(A) P(B)

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Example: Independence

Are the following events independent or dependent?

1) Winning at the Hoosier (or any other) lottery.2) The marching band is holding a raffle at a

football game with two prizes. After the first ticket is pulled out and the winner determined, the ticket is taped to the prize. The next ticket is pulled out to determine the winner of the second prize.

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Example: Independence

1. Deal two cards without replacementA = 1st card is a heart B = 2nd card is a heartC = 2nd card is a club.a) Are A and B independent?b) Are A and C independent?

2. Repeat 1) with replacement.

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Disjoint vs. Independent

In each situation, are the following two events a) disjoint and/or b) independent?1) Draw 1 card from a deck

A = card is a heart B = card is not a heart2) Toss 2 coins

A = Coin 1 is a head B = Coin 2 is a head3) Roll two 4-sided dice. A = red die is 2 B = sum of the dice is 3

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Example: Complex Multiplication Rule (1)The following circuit is in a series. The current

will flow only if all of the lights work. Whether a light works is independent of all of the other lights. If the probability that A will work is 0.8, P(B) = 0.85 and P(C) = 0.95, what is the probability that the current will flow?

http://www.berkeleypoint.com/learning/parallel_circuit.htmlA B C

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Example: Complex Multiplication Rule (2)The following circuit to the right is

parallel. The current will flow if at least one of the lights work. Whether a light works is independent of all of the other lights. If the probability that A will work is 0.8, P(B) = 0.85 and P(C) = 0.95, what is the probability that the current will flow?

http://www.berkeleypoint.com/learning/parallel_circuit.html

A

B

C

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Example: Complex Multiplication Rule (3)A diagnostic test for a certain disease has a

specificity of 95%. The specificity is the same as true negative, that is the test is negative when the person doesn’t have the disease.

a) What is the probability that one person has a false positive (the test is positive when they don’t have the disease)?

b) What is the probability that there is at least one false positive when 50 people who don’t have the disease are tested?

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4.3: Random Variables - Goals• Be able to define what a random variable is.• Describe the probability distribution of a discrete

random variable.• Use the distribution of a discrete random variable to

calculate probabilities of events.• Describe the probability distribution of a continuous

random variable.• Use the distribution of a continuous random

variable to calculate probabilities of events.

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Random Variables

• A random variable takes numerical values that describe the outcomes of some chance process.

• The probability distribution of a random variable gives its possible values and their probabilities.

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Discrete Random Variables

• A discrete random variable X takes a fixed set of possible values with gaps between.

• They are usually displayed in table form

• These probabilities must satisfy the following:1. 0 ≤ pi ≤ 1

2. Sum of all the pi’s is 1

value x1 x2 …probability p1 p2 …

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Examples: Probability Histograms

1 2 3 4-0.1999999999999996.10622663543836E-16

0.2000000000000010.4000000000000010.600000000000001

#1

Outcomes

Prob

abili

ty

1 2 3 40

0.20.40.6

#2

Outcomes

Prob

abili

ty

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Example: Discrete Random Variable

In a standard deck of cards, we want to know the probability of drawing a certain number of spades when we draw 3 cards. Let X be the number of spades that we draw.a) What is the distribution?b) What is the probability that you draw at least

1 spade?c) What is the probability that you draw at least

2 spades?

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Example: Discrete (cont.)

0 1 2 3-0.1

0.1

0.3

0.5

Spades Example

Number of Spades

Prob

abili

ty

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Normal Distribution: Example

A particular rash has shown up in an elementary school. It has been determined that the length of time that the rash will last is normally distributed with mean 6 days and standard deviation 1.5 days.

a) What is the percentage of students that have the rash for longer than 8 days?

b) What is the percentage of students that the rash will last between 3.7 and 8 days?

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4.4: Means and Variances of Random Variables - Goals

• Be able to use a probability distribution to find the mean of a discrete (or continuous) random variable.

• Be able to use the law of large numbers to describe the behavior of the sample mean.

• Calculate means using the rules for means.• Be able to use a probability distribution to find the

variance of a discrete (or continuous) random variable.

• Calculate variances using the rules for variances for both correlated and uncorrelated random variables.

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Formulas for the Mean of a Random Variable

• Discrete

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Example: Expected value

What is the expected value of the following:a) A fair 4-sided die

X 1 2 3 4

Probability 0.25 0.25 0.25 0.25

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Statistical Estimation

What would happen if we took many samples?

Population

Sample

Sample

Sample

Sample

Sample

Sample

Sample

Sample

?

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Law of Large Numbers

• Draw independent observations at random from any population with finite mean µ. The law of large numbers says that, as the number of observations drawn increases, the sample mean of the observed values gets closer and closer to the mean µ of the population.

• Our intuition doesn’t do a good job of distinguishing random behavior from systematic influences. This is also true when we look at data.

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Rules for MeansRule 1: If X is a random variable and a and b are

fixed numbers, then:µa+bX = a + bµX

Rule 2: If X and Y are random variables, then:µXY = µX µY

Rule 3: If X is a random variable and g is a function of X, then:

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Example: Expected ValueAn individual who has automobile insurance form a certain company is randomly selected. Let X be the number of moving violations for which the individual was cited during the last 3 years. The distribution of X is

a) Verify that E(X) = 0.60. b) If the cost of insurance depends on the following

function of accidents, g(y) = 400 + (100y -15), what is the expected value of the cost of the insurance?

X 0 1 2 3px 0.60 0.25 0.10 0.05

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Example: Expected ValueFive individuals who have automobile insurance from a certain company are randomly selected. Let X and Y be two different accident profiles in this insurance company:

E(X) = 0.60

E(Y) = 0.95 c) What is the expected value the total number of accidents

of the people if 2 of them have the distribution in X and 3 have the distribution in Y?

X 0 1 2 3px 0.60 0.25 0.10 0.05

Y 0 1 2 3pY 0.40 0.35 0.15 0.10

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Example: Expected valueAn individual who has automobile insurance form a certain company is randomly selected. Let X be the number of moving violations for which the individual was cited during the last 3 years. The distribution of X is

d) Calculate E(X2).

X 0 1 2 3px 0.60 0.25 0.10 0.05

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Variance of a Random Variable

= E(X2) – (E(X))2

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Example: Variance

An individual who has automobile insurance form a certain company is randomly selected. Let X be the number of moving violations for which the individual was cited during the last 3 years. The distribution of X is

e) Calculate Var(X).

X 0 1 2 3px 0.60 0.25 0.10 0.05

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Rules for VarianceRule 1: If X is a random variable and a and b are

fixed numbers, then:σ2

a+bX = b2σ2X

Rule 2: If X and Y are independent random variables, then:

σ2XY = σ2

X + σ2Y

Rule 3: If X and Y have correlation ρ, then:

σ2XY = σ2

X + σ2Y 2ρσXσY

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Correlation

i i

X Y

(x x)(y y)r

n 1 s s

i X i Y X,Y

X Y

(x )(y )p

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Example: VarianceAn individual who has automobile insurance form a certain company is randomly selected. Let X be the number of moving violations for which the individual was cited during the last 3 years. The distribution of X is

a) Calculate the variance of this distribution. b) If the cost of insurance depends on the following

function of accidents, g(y) = 400 + (100y -15), what is the standard deviation of the cost of the insurance?

X 0 1 2 3px 0.60 0.25 0.10 0.05

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Example: Variance5 individuals who have automobile insurance from a certain company are randomly selected. Let X and Y be two different independent accident profiles in this insurance company:

Var(X) = 0.74

Var(Y) = 0.95 What is the standard deviation of the difference between

the 2 who have insurance using the X distribution and the 3 who have insurance using the Y distribution?

X 0 1 2 3px 0.60 0.25 0.10 0.05Y 0 1 2 3pY 0.40 0.35 0.15 0.10

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4.5: General Probability Rules - Goals• Apply the five rules of probability (again).• Apply the generation addition rule.• Be able to calculate conditional probabilities.• Apply the general multiplication rule.• Be able to use tree diagram.• Use Bayes’s rule to find probabilities.• Determine if two events with positive probability

are independent.

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Probability RulesRule 1. The probability P(A) of any event A

satisfies 0 ≤ P(A) ≤ 1.Rule 2. If S is the sample space in a probability

model, then P(S) = 1.Rule 3. If A and B are disjoint,

P(A or B) = P(A) + P(B). Rule 4: For any event A, P(AC) = 1 – P(A).Rule 5: If A and B are independent:

P(A and B) = P(A) P(B)

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General Addition Rule

P(A or B) = P(A) + P(B) – P(A and B)Select a card at random from a deck of cards.

What is the probability that the card is either an Ace or a Heart?

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Example: Venn Diagrams

At a certain University, the probability that a student is a math major is 0.25 and the probability that a student is a computer science major is 0.31. In addition, the probability that a student is a math major and a student science major is 0.15.

a) What is the probability that a student is a math major or a computer science major?

b) What is the probability that a student is a computer science major but is NOT a math major?

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Conditional Probability

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Conditional Probability: ExampleA news magazine publishes three columns entitled

"Art" (A), "Books" (B) and "Cinema" (C). Reading habits of a randomly selected reader with respect to these columns are

a) What is the probability that a reader reads the Art column given that they also read the Books column?

b) What is the probability that a reader reads the Books column given that they also read the Art column?

Read Regularly

A B C A and B A and C B and C

Probability 0.14 0.23 0.37 0.08 0.09 0.13

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Example: General Multiplication Rule

Suppose that 8 good and 2 defective fuses have been mixed up. To find the defective fuses we need to test them one-by-one, at random. Once we test a fuse, we set it aside.

a) What is the probability that we find both of the defective fuses in the first two tests?

b) What is the probability that when testing 3 of the fuses, the first tested fuse is good and the last two tested are defective?

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Independence RevisitedGeneral multiplication rule:

P(A and B) = P(A) P(B|A)If A and B are independent:

P(A and B) = P(A) P(B)Therefore, if A and B are independent:

P(B|A) = P(B)

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Example: Tree Diagram/Bayes’s RuleA diagnostic test for a certain disease has a 99%

sensitivity and a 95% specificity. Only 1% of the population has the disease in question. If the diagnostic test reports that a person chosen at random from the population tests positive, what is the probability that the person does, in fact, have the disease?

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Bayes’s Rule• Suppose that a sample space is decomposed

into k disjoint events A1, A2, … , Ak —none of which has a 0 probability—such that

• Let B be any other event such that P(B) is not 0. Then

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Law of Total Probability

B and 3

2

34 5

6

7

1

B and 4 B and 6

B and 7

B