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Learn to let go. That is the key to happiness.~Jack Kornfield

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Probability

Section 4.1-4.5Basic terms and rulesConditional probability and independenceBayes’ rule

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This lady has lost 10 games in a row on this slot machine. Would you play this slot machine or another one?

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Random Circumstance

A random circumstance is one in which the outcome is unpredictable. The outcome is unknown until we observe it.

Eg. Toss a die

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Interpretations of Probability

Given a random circumstance, the probability of a specific outcome can be interpreted as

1. (classical interpretation) a % arising from the nature of the circumstance.

2. (relative frequency interpretation) the proportion of times this outcome will occur over a large number of the same circumstances.

3. (personal interpretation) what a person believes.

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Two Ways to Determine Probability

Making an assumption about the physical world and use it to find the probabilities.

Repeating the same circumstances many times and calculating the relative frequencies.

Based on the person’s experiences.

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Example 1: Coin Flipping

What is the probability that a flipped coin shows heads up?

Simulation in Minitab (random digit table and binomial way)

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Example 2: Traffic Jam on the I-880

< = 30 mph traffic jam

How often will a driver encounter traffic jam on the I-880 during 7-9 am in weekdays?

Ans: __ out of 100 times.

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Terms

Sample space SS: the collection of all possible outcomes of a random circumstance

An event is a collection of one or more outcomes in the sample space.

A simple event is an event of one outcome.

** what are the random circumstances of the examples?** what are the outcomes/sample space/event of the examples?

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Probability Models

A probability model assigns a value to each outcome which satisfies the following properties:The probability of an outcome must be between 0 and 1

The sum of the probabilities over all possible outcomes must be 1 (i.e. P(SS) =1)

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Probability of an Event

The sum of the probabilities of outcomes in the event

** Revisit the examples: 1) build up probability models and 2) find probabilities of events

Probability Models

A probability model with a finite sample space is called discrete

A continuous probability model assigns probabilities as areas under a density curve. The area under the curve and above any range of values is the probability of an outcome in that range

Example: randomly pick a number between 0 and 1

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Equally Likely Probability Model

If the sample space S is finite in number and the outcomes have the same likelihood of occurrence, then each outcome has probability equal to 1 divided by the number of possible outcomes and so

.Sin #

Ain #)( AP

Random Variables

A random variable is a function assigning a real value to an outcome of a RC

The distribution of a random variable X tells us what values X can take and how to assign probabilities to those values

Example:

1. # of dots (RC: rolling a die)

2. height of a student (RC: randomly pick from the class)

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Basic Event Relations

Mutually exclusive eventsTwo events A and B are called mutually exclusive if the occurrence of one excludes the occurrence of the other.

Complement eventsThe complement of an event A is the event that A does not occur, denoted as A.

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Basic Probability Laws

The union of 2 events A and B, , is the event when either A or B or both occur.

The intersection of A and B, , is the event when A and B both occur.

Venn diagram shows us that

BA

BA

)()()()( BAPBPAPBAP

Operation Laws

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Conditional Probability

Base on a survey of 1000 government employees:

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Conditional Probability

1. If an employee is selected randomly (out of the 1000 surveyed), what is the probability that the selected one is a male employee?

2. If a male employee is selected, what is the probability that he is also married? (called the conditional probability of selecting a married employee given that the selected one is male.)

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Conditional Probability

P(married| male)=

# of married male employees

# of male employees

i.e. proportion of married male employees

proportion of male employees

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Conditional Probability

In general, the conditional probability of event B given event A is

)(

)()|(

BP

BAPBAP

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Independent Events

Event A is called independent of event B if

the knowledge that B has occurred

DOES NOT

change the probability of the occurrence of A,

i.e. P(A|B) = P(A).

Does P(B|A) = P(B)?

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Independent Events

Events A and B are independent events if and only if, P(A|B) = P(A) or P(B|A) = P(B).

Otherwise, A and B are dependent. A and B are independent if and only if,

A1, A2, …, Ak are independent if and only if,

).()()( BPAPBAP

).()...2()1()...21( AkPAPAPAkAAP

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Independent Events

An event cannot be both mutually exclusive and independent (unless it is trivial i.e. probability 0):

If events are independent, then they cannot be mutually exclusive.

If events are mutually exclusive, they cannot be independent.

Independent Events

Example 1:

A red die and a white die are rolled. Define the events:

A= 4 on red die; B= sum of two dice is odd.

Show that A and B are independent.

Example 2:

Given that P(grade A in 6204)= .60; P(grade A in 6304)= .60; P(grade A in both) =.36. Are A, B independent?

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Diagnostic Tests

A diagnostic testing or screening is the application of a test to individuals who have not yet exhibited any clinical symptoms in order to classify them with respect to their probability of having a particular disease.

“sensitivity” is the true + rate “specificity” is the true - rate “prevalence” is the proportion of subjects with

the disease in a population26

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Diagnostic Tests

Consider a common pregnancy test

Truth\Test result Positive Negative

Pregnancy Correct False Negative

No pregnancy False Positive Correct

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Diagnostic Tests

Eg. Pregnancy tests

Sensitivity = P( + | pregnancy)

False positive rate = P( + | non-preg)

Specificity = P( - | non-preg)

False negative rate = P( - | pregnancy)

Q: What is the probability that a woman with a positive result is actually NOT pregnant?

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Bayes’ Rule

)()|()()|(

)()(

)()|(

APABPAPABP

BAPBP

BAPBAP

Example: Pap Smear

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Bayes’ Theorem

)()|(...)1()1|(

)()|(

AkPAkBPAPABP

BAiPBAiP

Think of the events A1, A2,…, Ak as representing all possible conditions that can produce the observable “effect” B. In this context, the probabilities P(Ai)’s are called prior probabilities. Now suppose that the effect B is observed to occur. Bayes’ theorem gives a way to calculate the probability that B was produced or caused by the particular condition Ai than by any of the other conditions. The conditional probability P(Ai|B) is called the posterior probability of Ai .

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Steps for Finding Probabilities

1. Identify random circumstance2. Identify the sample space3. Assign whatever probabilities you know

(building a probability model if possible)4. Specify the event for which the probability is

wanted5. Use the probabilities from step 3 and the

probability rules to find the probability of interest

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Example 4.3:

A book club classifies members as heavy, medium, or light purchasers, and separate mailings are prepared for each of these groups. Overall, 20% of the members are heavy purchasers, 30% medium, and 50% light.

Example 4.3:

The following % are obtained from existing records of individuals classified as heavy, medium, or light purchasers:

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1st 3 months’ purchases

Group (%)

Heavy Medium Light

0 5 15 60

1 10 30 20

2 30 40 15

3+ 55 15 5

Example 4.3:

If a member is a heavy purchaser, what is the probability he/she will buy 2 books in the first 3 months?

What percent of members will buy 2 books in the first 3 months?

If a member purchases 2 books in the first 3 months, what is the probability that he/she is a light purchaser?

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Tools for Finding Probabilities

When conditional or joint probabilities are known for two events Two-way tables

For a sequence of events, when conditional probabilities for events based on previous events are known Tree diagrams

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Example:

People are classified into 8 types. For instance, Type 1 is “Rationalist” and applies to 15% of men and 8% of women. Type 2 is “Teacher” and applies to 12% of men and 14% of women. Each person fits one and only one type.

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Suppose college roommates have a particularly hard time getting along with each other if they are both “Rationalists.”A college randomly assigns roommates of the same sex.

What proportion of male roommate pairs will have this problem?

What proportion of female roommate pairs will have this problem?

Assuming that half of college roommate pairs are male and half are female. What proportion of all roommate pairs will have this problem?

A psychologist has noticed that “Teachers” and “Rationalists” get along particularly well with each other, and she thinks they tend to marry each other.

One of her colleagues disagrees and thinks that the “types” of spouses are independent of each other.

• If the “types” are independent, what is the probability that a randomly selected married couple would consist of one “Rationalist” and one “Teacher”?

• In surveys of thousands of randomly selected married couples, she has found that about 5% of them have one “Rationalist” and one “Teacher.” Does this contradict her colleague’s theory that the types of spouses are independent of each other?