Lecture 05. Introduction to Probability Web

7/28/2019 Lecture 05. Introduction to Probability Web

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Statistics

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ST 361: Introduction to Statistics

Introduction to Probability

Kimberly Weems

[email protected]

5260 SAS Hall


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Outline

Probability Trees Probability Models

Sample Spaces, Events, Venn Diagrams

Axioms of Probability Probability Rules (Laws)

Addition Rule

Multiplication Rule


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Example: Southwest Energy

A Southwest Energy Company pipeline has 3 safety shutoffvalves in case the line starts to leak.

The valves are designed to operate independently of one

another:

7% chance that valve 1 will fail 10% chance that valve 2 will fail

5% chance that valve 3 will fail

If there is a leak in the line, find the following probabilities:

a. That all three valves operate correctly

b. That all three valves fail

c. That only one valve operates correctly

d. That at least one valve operates correctly


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Probability Tree Approach

A probability tree is a useful way to visualize

this problem and to find the desiredprobability.


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A: P(all three valves operate correctly)

P(all three valves work)

= .93*.90*.95

= .79515


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B: P(all three valves fail)

P(all three valves fail)

= .07*.10*.05

= .00035


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D: P(at least one valve operates correctly)

P(at least one valve operatescorrectly

= 1P(no valves operate correctly)

= 1 - .00035 = .99965

7 paths

1 path


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Example: AIDS Testing

V={person has HIV}; CDC: P(V)=.006

+: test outcome is positive (test indicates

HIV present)

-: test outcome is negative

clinical reliabilities for a new HIV test:

1. If a person has the virus, the test result will be

positive with probability .999

2. If a person does not have the virus, the test result

will be negative with probability .990


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Question 1

What is the probability that a randomlyselected person will test positive?


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Probability Tree

clinical

reliability

clinical

reliability


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Probability Tree

Multiply

branch probsclinical

reliability

clinical

reliability


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Question 1 Answer

What is the probability that a randomlyselected person will test positive?

P(+) = .00599 + .00994 = .01593


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Question 2

If your test comes back positive, what is theprobability that you have HIV?

(Remember: we know that if a person has thevirus, the test result will be positive with

probability .999; if a person does not have thevirus, the test result will be negative withprobability .990).

Looks very reliable


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Question 2 Answer

Answertwo sequences of branches lead to positive test;

only 1 sequence represented people who have

HIV.P(person has HIV given that test is positive)

=.00599/(.00599+.00994) = .376


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Summary

Question 1:

P(+) = .00599 + .00994 = .01593

Question 2: two sequences of branches lead to

positive test; only 1 sequence representedpeople who have HIV.

P(person has HIV given that test is positive)

=.00599/(.00599+.00994) = .376


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Recap

We have a test with very high clinical reliabilities:1. If a person has the virus, the test result will be positivewith probability .999

2. If a person does not have the virus, the test result will benegative with probability .990

But we have extremely poor performance when thetest is positive:

P(person has HIV given that test is positive) =.376

In other words, 62.4% of the positives are falsepositives! Why?

When the characteristic the test is looking for israre, most positives will be false.


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Probability models

Aprobability modelis a mathematical representationof a random phenomenon. It is defined by its

sample space,

events within the sample space, and

probabilities associated with each event.


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Sample Space

Random experiments have unique outcomes.

The set of all possible outcome of a random experiment is

called the sample space, S.

Sis discrete if it consists of a finite or countable infinite set ofoutcomes.

Sis continuous if it contains an interval (either a finite or

infinite width) of real numbers.


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Example: Sample Spaces

[Sis continuous] Randomly select and measure the thickness

of a part.

S=R+ = {x|x > 0}, the positive real line. Negative or zero

thickness is not possible.

[Sis continuous, finite width] It is known that the thickness is

between 10 and 11 mm. We have S= {x|10


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Example: Sample Spaces (contd)

[Sis continuous] Two parts are randomly selected & measured.

S=R+ xR+, Sis continuous.

[Sis discrete] Do the 2 parts conform to specifications?

S= {yy, yn, ny, nn}, Sis discrete.

[Sis discrete] Number of conforming parts?

S= {0, 1, 2}, Sis discrete.

[Sis discrete, countable infinite ] Parts are randomly selected

until a non-conforming part is found.

S= {n, yn, yyn, yyyn, },

Sis countably infinite.

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Events Are Sets of Outcomes

An event (E) is a subset of the sample space of a random

experiment, i.e., one or more outcomes of the sample space.

Event combinations are:

Union of 2 events = the event consisting of all outcomes

that are contained in either of two events, E1

U E2. Called

E1 or E2.

Intersection of 2 events = the event consisting of all

outcomes that contained in both of two events, E1 E2.

Called E1 and E2.

Complement of an event = the set of outcomes that are not

contained in the event, E ornot E, or Ec .

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Example: Discrete Event Algebra

Recall the sample space from Example 2, S= {yy, yn, ny, nn}

concerning conformance to specifications.

Let E1 denote the event that at least one part does conform

to specifications, E1 = {yy, yn, ny}

Let E2 denote the event that no part conforms to

specifications, E2 = {nn}

Let E3 = , the null or empty set.

Let E4 = S, the universal set.

Let E5

= {yn, ny, nn}, at least one part does not conform.

Then E1 U E5 = S

Then E1 E5 = {yn, ny}

Then E1= {nn}

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Example: Continuous Event Algebra

Measurements of the thickness of a part are modeled with the

sample space: S= R+.

Let E1 = {x|10 x < 12}, show on the real line below.

Let E2 = {x|11


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Venn Diagrams Show Event Relationships

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Events A & B contain their respective outcomes. The shaded

regions indicate the event relation of each diagram.

Venn diagrams

i f ll l i


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Venn Diagram of Mutually Exclusive

Events

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Events A & B are mutually exclusive because they share no

common outcomes.

The occurrence of one event precludes the occurrence of the

other.

Symbolically, A B =

Mutually exclusive events


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What is Probability?

Probability is the likelihood or chance that a particular

outcome or event from a random experiment will occur.

Here, only finite sample spaces ideas apply.

Probability is a number in the [0,1] interval.

May be expressed as a:

proportion (0.15)

percent (15%)

fraction (3/20)

Generally speaking, a probability of: 1 indicates highly likely

0 indicates highly unlikely

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P b bili B d E ll Lik l


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Probability Based on Equally-Likely

Outcomes Whenever a sample space consists ofNpossible outcomes that

are equally likely, the probability of each outcome is 1/N.

Example: Consider an unbiased die with 10 faces labeled

1,2,10. Unbiased means that no face is favored, when

throwing the die; thus each face has an equal chance of being

shown. We throw the die; the probability that the die shows

face 10 is 1/10 or 0.1, because each outcome in the sample

space is equally likely.

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Probability of an Event

For a discrete sample space, theprobability of an event E,

denoted byP(E), equals the sum of the probabilities of the

outcomes inE.

The discrete sample space may be:

A finite set of outcomes

A countably infinite set of outcomes.

Further explanation is necessary to describe probability with

respect to continuous sample spaces.

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Equally likely outcomes (uniform model)

Equally likely outcomes (uniform model):

If a random phenomenon had kpossible outcomes, all equally

likely , then each individual outcome has probability 1/k. The

probability of any event A is:

P(A) = {count of outcomes in A} / {count of outcomes in S}

= |A| / k.


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Example: Probabilities of Events

A random experiment has a sample space {w,x,y,z}. These

outcomes are not equally-likely; their probabilities are: 0.1,

0.3, 0.5, 0.1.

Event A ={w,x}, event B = {x,y,z}, event C = {z}

P(A) = _______

P(B) = _______

P(C) = _______

P(A) = _____ and P(B) = ____ and P(C) = ______

Since event AB = {x}, then P(AB) = _____ Since event AUB = {w,x,y,z}, then P(AUB) = ______

Since event AC = {null}, then P(AC ) = ________

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Axioms of Probability

Probability is a number that is assigned to each member of a

collection of events from a random experiment that satisfies

the following properties:

1. P(S) = 1

2. 0 P(E) 1, for any event E

3. For each two eventsE1andE2 withE1E2 = ,

P(E1UE2) =P(E1) +P(E2)addition rule

These imply that: P() =0 ;P(E) = 1P(E)complement rule

IfE1is contained inE2, thenP(E1)P(E2).

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Probability Rules

Joint events are generated by applying basic set operations to

individual events, specifically:

Unions of events,A UB

Intersections of events,AB

Complements of events,A

Probabilities of joint events can often be determined from the

probabilities of the individual events that comprise it. And

conversely.

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Birthday Problem

What is the smallest number of people youneed in a group so that the probability of2 or

morepeople having the same birthday is

greater than 1/2?

Answer: 23

No. of people 23 30 40 60

Probability .507 .706 .891 .994


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Birthday Problem

A={at least 2 people in the group have acommon birthday}

A = {no one has common birthday}

502.498.1)'(1)(

498.365

343

365

363

365

364)'(

:23

365

363

365

364)'(:3

APAPso

AP

people

APpeople


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Probability rules

Addition Rule for Mutually Exclusive Events: Recall that

two events A and B are mutually exclusive (or disjoint) events

if they have no outcomes from S in common

If A and B are mutually exclusive events, then

P(A or B) = P(A) + P(B)


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Probability rules

General Addition Rule: For any two events A and B

P(A or B) = P(A) + P(B)P(A and B)


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Probability rules

Multiplication Rule for independent events. Two events, A

and B are independent if the occurrence of one does not affectthe probability that the other one will occur.

If A and B are independent then

P(A andB) = P(A) P(B).

Remark on independence: The fact that a coin tossed with my

left hand comes up T rather than H, does not influence the

outcome of a coin tossed with my right hand.

The probability of falling on the street is NOT independent of

whether it has snowed. These events are dependent.


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St ti ti

Some examples

Example1: We roll a 6-sided die.

The sample space (set of all possible outcomes) S = ____

The simple events are:___________.

The event A that the outcome is {3} is _______.

Example 2. We roll a die, and the event of interest,E, is

obtaining an odd number. That is . What is the

probability of this event ?

Let . What is the probability ofF?

1,3,5E

2,4,6F

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Lecture 05. Introduction to Probability Web