math230 probability

8/20/2019 MATH230 probability

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TEDU, Fall 2015

TEDU

MATH 230

Introduction to ProbabilityTheory


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Introduction.

2.1. Sample space.

2.2. Events.

2.3 Counting sample points.

2.4 Probability of an event.2.5 Additive Rules.

2.6 Conditional probability, Independence, and Product rule.

2.7 Bayes’ rule.

Chapter 2 - Probability


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introduction

Probability is a fraction expressing the chance that a certain

event will occur.

allows you to handle uncertainty

forms a major component that supplements statistical methods

What does it mean if an event has a probability of 1/3rd of

occurring?

If the experiment is repeated a large number of times, the

event will occur 1/3rd

of the time.


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Definition 2.1

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Sample Space

Experiment: Any process that generates a set of data.

Outcome: The result of an experiment

Random Experiment: An experiment whose outcome is not

known in advance.

e.g: tossing of a coin only two possible outcomes: H or T

Trial: Each observation of an experiment


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Tree diagram for Example


An experiment consists of:

Flipping a coin and thenflipping it a second time if a

head occurs.

If a tail occurs on the first flip,

then a die is tossed once.

Example 2.2


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Experiment:

Three items are selected

at random.

Each item is classified

defective (D), or

nondefective (N).

Example 2.3


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Definition 2.2


Events

For any given experiment, we may be interested in the

occurrence of a certain event.

Event A (Example-2.3): The number of defectives is smaller than 2.A={DNN,NDN,NND,NNN}


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Definition 2.3


Events

Event A : The number of defectives is smaller than 2.

A={DNN,NDN,NND,NNN}

A’={DDD, DDN, DND, NDD}


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Definition 2.4


Events

Definition 2.6

Definition 2.5


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Exercise (2.5 and 2.9, page 42-43)


An experiment consists of tossing a die and then flipping a coin once if

the number on the die is even. If the number on the die is odd, the coin

is flipped twice.

Using the notation 4H, for example, to denote the outcome that the die

comes up 4 and then the coin comes up heads, and 3HT to denote theoutcome that comes up 3 followed by a head and then a tail on the coin: a) List the elements in sample space.

b) List the elements corresponding to the event A that a number less

than 3 occurs on the die.c) List the elements corresponding to the event B that two tails occur.

d) List the elements corresponding to the event A’

e) List the elements corresponding to the event A’ ∩ B

f) List the elements corresponding to the event A U B


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Exercise (2.5 and 2.9, page 42-43)

Venn Diagram


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The relationship between events and the corresponding

sample space can be illustrated graphically by meansof Venn diagrams.

We let the sample space be a rectangle and represent

events by circles drawn inside the rectangle.

Events - Venn Diagrams


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Events represented by various regions

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Addison-Wesley. All rights

reserved.

Events - Venn Diagrams


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Experiment: Select a card atrandom from an ordinary deck of

52 cards.

Events:

A: The card is red.B: The card is jack, queen or king of

the diamands.

C: The card is an ace.

Are there any mutually exclusive

(disjoint) events ?

A ∩ B = ?

B ∩ C = ?

A U B = ?

Events of the Sample Space


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Events

Several results that follow from the foregoing

definitions, which may easily verified by means of Venn

diagrams, are as follows:

1. A = ?

2. A = ?

3. A A’ = ?

4. A A’ = ? 5. S’ = ?

6.’ = ?

7. (A’)’ = ?

8. (A B)’ = ?

9. (A B)’ = ?


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Events


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Example: In a medical study, patients

are classified in 8 ways according to

whether they have blood type AB+,

AB

-

, A

+

, A

-

, B

+

, B

-

, 0

+

, or 0

-

and alsoaccording to whether their blood

pressure is low, medium, or high.

Find the number of ways in which a

patient can be classified.

Counting Sample Points

8 * 3 = 24 ways


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Rule 2.1(Multiplication Rule)




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Example: A developer of new subdivision offers a

prospective home buyer a choice of 4 designs, 3different heating systems, a garage or carport, and a

balcony and screened porch. How many different plans

are available to this buyer?



n1 * n2 * n3 * n4 = 4 * 3 * 2 * 2 = 48 ways


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Rule 2.2 (Generalized Multiplication Rule)

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reserved.



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Definition 2.7



Consider the three letters a, b, and c. There are 6 distinct

permutations: abc, acb, bac, bca, cab, and cba. Using Rule 2.2,we could arrive at the answer 6 without actually listing the

different orders:

There are n1=3 choices for the first position.

No matter which letter is chosen, there are always n2=2

choices for the second position. No matter which two letters are chosen for the first two

positions, there is only n3=1 choice for the last position, giving a

total of n1n2n3 = (3)(2)(1) = 6 permutations.


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The number of distinct arrangements of n objects? n objects can go in the first position.

Once the first object is fixed, n-1 objects can go in the second position.

Then n-2 objects in the third position, etc.

Number of arrangements is n(n-1)(n-2) … (1).

Definition 2.8

Theorem 2.1


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Counting Sample Points(Example 2.38, Page 52)


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Theorem 2.2



Exercise 2.41, Page 52. Find the number of ways that 6

teachers can be assigned to 4 sections of an introductory

psychology course if no teacher is assigned to more than one

section.


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Theorem 2.3

Example: If 3 people are playing with decks,we do not have a new permutation if they all move

one position in the same direction. By considering

one person is fixed position and arranging theother two in 2! ways, we find 2 distinct

arrangements.


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So far we have considered permutations of distinct

objects.

Obviously, if the letters b and c are both equal to x,then the 6 permutations of the letters a, b, and c

become axx, axx, xax, xax, xxa, and xxa, of which

only 3 are distinct.

Therefore, with 3 letters, 2 being the same, we have

3!/2! = 3 distinct permutations.


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Theorem 2.4

How many distinct permutations can be made

from the letters of the word INFINITY ?

Counting Sample Points(Exercise 2.45, Page 52)


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Theorem 2.5

In how many ways can 7 graduate students be assigned to 1

triple and 2 double hotel rooms during a conference?


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In many problems, we are interested in the number of ways of

selecting r objects from n without regard to order.

These selections are called combinations.

A combination is actually a partition with two cells, the one cellcontaining the r objects selected and the other cell containing the

(n−r) objects that are left. The number of such combinations,denoted by:

is usually shortened to

since the number of elements in the second cell must be n − r.


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Theorem 2.6


Exercise 2.48 (Page 52). How many ways are there to select 3

candidates from 8 equally qualified recent graduates for

openings in an accounting firm?



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Probability of an Event

The likelihood of the occurrence of an event is evaluated by

means of a set of real numbers, called weights or probabilities,

ranging from 0 to 1.

To every point in the sample space we assign a probability such

that the sum of all probabilities is 1.

If we have reason to believe that a certain sample point is quite

likely to occur when the experiment is conducted, the probability

assigned should be close to 1.

On the other hand, a probability is closer to 0 is assigned to a

sample point that is not likely to occur.


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Definition 2.9




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1. Logical Probabilities: In experiments having a certain

symmetry, we often have equally likely outcomes.-Tossing a coin, P(H) = P(T ) = 1/2;- Rolling a die, P(1) = P(2) = ... = P(6) = 1/6;

2. Relative Occurence based Probabilities: If we can repeat

the experiment a large number of times,

Ex. P(number of customers / day arriving at a bank > 100).

3. Subjective Probabilities: P(A) is assigned based on judgment

when the experiment cannot be repeated,

e.g. P(earthquake will occur in Turkey in five years).



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Rule 2.3




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Exercise 2.58, Page 60. A pair of fair dice is tossed. Find the

probability of getting

a) a total of 8;

b) at most a total of 5.

Additive Rules


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Exercise 2.60, Page 60. If 3 books are picked at random from a

shelf containing 5 novels, 3 books of poems, and a dictionary,

what is the probability that

a) the dictionary is selected?b) 2 novels and 1 book of poems are selected?

Additive Rules


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Theorem 2.7


Additive Rules


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Corollary 2.1


Corollary 2.2

Corollary 2.3

Additive Rules


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Theorem 2.8


Theorem 2.9

Additive Rules


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Exercise 2.56, Page 60. An automobile manufacturer is

concerned about a possible recall of its best-selling four-door

sedan. If there were a recall, there is a probability of 0.25 of a

defect in the brake system, 0.18 of a defect in the transmission,0.17 of a defect in the fuel system, and 0.40 of a defect in some

other area.

a) What is the probability that the defect is the brakes or the

fueling system if the probability of defects in both systemssimultaneously is 0.15?

b) What is the probability that there are no defects in either the

brakes or the fueling system?

Additive Rules


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Exercise 2.56, Page 60.

Additive Rules

C diti l P b bilit


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Conditional Probability,

Independence, and Product Rule

The probability of an event B occurring when it isknown that some event A has occurred is called a

conditional probability and is denoted by P(B|A).

C diti l P b bilit


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Definition 2.10




C diti l P b bilit


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Suppose that our sample space S is the population of adults in a smalltown who have completed the requirements for a college degree.

We shall categorize them according to gender and employment status:

One of these individuals is to be selected at random for a tour throughout

the country to publicize the advantages of establishing new industries inthe town. We shall be concerned with the following events:

M: a man is chosen,E: the one chosen is employed.




Conditional Probability


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Categorization of the Adults in a Small Town




What is the probability that the one chosen is employed?

P(E)=600/900=2/3

What is the probability that a male is chosen?P(M)=500/900=5/9

What is the probability that one chosen is employed and

male ? P(E M) =460/900=23/45



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Note that these probabilities do not reflect the

“association” between being a women and unemployed,

e.g., what percentage of the men are employed?

what percentage of the people who are employed

are men?

To investigate the association, do conditioning.





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Categorization of the Adults in a Small Town




What is the probability that the one chosen is employed

given that a man is chosen?

P(E|M)=P(E∩M)/P(M)=460/500=23/25 What is the probability that a man is chosen given that

the one chosen is employed?

P(M|E)= P(E ∩ M)/P(E)= 460/600=23/30.



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Example 2.37: One bag contains 4 white balls and 3 black balls,and a second bag contains 3 white balls and 5 black balls. One

ball is drawn from the first bag and placed unseen in the second

bag.

What is the probability that a ball now drawn from the second bag

is black?





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Definition 2.11


Theorem 2.10



Theorem 2.11



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An electrical system for Example 2.39 The system works ifcomponents A and B work and either of the components C or Dworks. The reliability (probability of working) of each component is

also shown. Find the probability that

(a) the entire system works and(b) the component C does not work, given that the entire systemworks. Assume that the four components work independently.





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Theorem 2.12




Definition 2.12


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Bayes’ Rule

Suppose that we are now given the additional information that

36 of those employed and 12 of those unemployed are members

of the Rotary Club.

Find the probability of the event A that the individual selected is a

member of the Rotary Club.

P(E)=600/900=2/3

P(E’)=300/900=1/3

P(A|E)=36/600=3/50P(A|E’)=12/300=1/25

P(A)=?


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Venn diagram for the events A, E and E ’


Bayes’ Rule

We can write A as the union of the two mutually exclusive events:

E∩A and E’∩A.

Hence, A = (E∩A)∪(E’∩A), we can write:

P(A) = P[(E ∩ A) ∪ (E’ ∩ A)] = P(E ∩ A) + P(E’ ∩ A)= P(E)P(A|E) + P(E’)P(A|E’).


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Bayes’ Rule

P(A) = P(E)P(A|E) + P(E’)P(A|E’).


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Theorem 2.13 – Theorem of Total Probability (Rule of

Elimination)


Bayes’ Rule


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Example 2.41, Page 74


Bayes’ Rule

In a certain assembly plant, three machines, B1, B2, and B3, make30%, 45%, and 25%, respectively, of the products. It is knownfrom past experience that 2%, 3%, and 2% of the products

made by each machine, respectively, are defective.

Now, suppose that a finished product is randomly selected. What

is the probability that it is defective?

Consider the following events

A : the product is defective,B1: the product is made by machine B1,

B2: the product is made by machine B2,

B3: the product is made by machine B3.


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Tree diagram for Example 2.41


Bayes’ Rule


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Theorem 2.14


Bayes’ Rule


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Recall Example 2.41, Page 74:


Bayes’ Rule

In a certain assembly plant, three machines, B1, B2, and B3, make30%, 45%, and 25%, respectively, of the products. It is known frompast experience that 2%, 3%, and 2% of the products made by each

machine, respectively, are defective.Now, suppose that a finished product is randomly selected and

detected as defective. What is the probability that it is made by B3?

3 3 | 33|

1 | 1 2 | 2 3 | 3

0.25 0.02 0.0050.204

0.3 0.02 0.45 0.03 0.25 0.02 0.0245

P B A P B P A B

P B AP A P B P A B P B P A B P B P A B


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Monty Hall Problem

Suppose you're on a game show, and you're given the

choice of three doors: Behind one door is a car;

behind the others, goats. You pick a door, say No. 1,

and the host, who knows what's behind the doors,opens another door, say No. 2, which has a goat. He

then says to you, "Do you want to pick door No. 3?" Is

it to your advantage to switch your choice?

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