06b Probability

8/4/2019 06b Probability

1/52

Probability

I Introduction to Probability

A Satisfactory outcomes vs. total outcomes

B Basic Properties

C Terminology

II Combinatory Probability

A The Addition Rule Or

1. The special addition rule (mutually exclusive events)

2. The general addition rule (non-mutually exclusive events)B The Multiplication Rule And

1. The special multiplication rule (for independent events)

2. The general multiplication rule (for non-independent events)


2/52

Probability for Equally Likely Outcomes

Suppose an experiment has N possible outcomes, all equally

likely. Then the probability that a specified event occurs equals

the number of ways,f, that the event can occur, divided by the

total number of possible outcomes. In symbols

Probability of a given event =N

f

Number of ways a given event can occur

Total of all possible outcomes


3/52

Frequency distribution of annual income for U.S.

families


4/52

Probability from Frequency Distributions

What is the a priori probability

of having an income between

$15,000 and $24,999


5/52

Frequency distribution for students ages

N= 40


6/52

Frequency distribution for students ages

What is the likelihood of randomly selecting a student who is

older than 20 but less than 22?

What is the likelihood of selecting a student whos age is an

odd number?

What is the likelihood of selecting a student who is either 21

or 23?


7/52

Sample space for rolling a die once


8/52

Possible outcomes for rolling a pair of dice


9/52

Probabilities of 2 throws of the die

What is the probability of a 1 and a 3?

What is the probability of two sixes? What is the probability of at least one 3?

2/36

1/3612/36


10/52

The Sum of Two Die TossesSum Frequency

2 13 2

4 35 4

6 57 6

8 5

9 4

10 311 2

12 1

What is the probability that the

sum will be

5?

7?


sum will be 10 or more?


sum will be either 3 or less or 11

or more?

4/36

6/36

6/36

3/36 + 3/36


11/52

Two computer simulations of tossing a balanced coin

100 times


12/52

Basic Properties of Probabilities

Property 1: The probability of an event is always between 0

and 1, inclusive.

Property 2: The probability of an event that cannot occur is 0.

(An event that cannot occur is called an impossible event.)

Property 3: The probability of an event that must occur is 1.

(An event that must occur is called acertain event.)


13/52

A deck of playing cards


14/52

The event the king of hearts is selected

1/52


15/52

The event a king is selected

1/13 = 4/52


16/52

The event a heart is selected

1/4 = 13/52


17/52

The event a face card is selected

3/13=13/52


18/52

Sample Space and Events

Sample space: The collection of all possible

outcomes for an experiment.

Event: A collection of outcomes for the

experiment, that is, any subset of the sample

space.


19/52

Probability Notation

IfEis an event, thenP(E) stands for the

probability that eventEoccurs. It is read theprobability of E


20/52

Venn diagram for eventE


21/52

Relationships Among Events

(notE): The event that Edoes not occur.

(A &B): The event that bothA andB occur.

(A orB): The event that eitherA orB or both

occur.


22/52

Event (notE) whereEis the probability of drawing a

face card.

40/52=10/13


23/52

An event and its complement


24/52

The Complementation Rule

For any eventE,

P(E) = 1 P (~E).

In words, the probability that an event occurs equals 1

minus the probability that it does not occur.


25/52

Combinations of Events

The Addition Rule Or

The special addition rule (mutually exclusive events)

The general addition rule (non-mutually exclusive events)

The Multiplication Rule And

The special multiplication rule (for independent events)

The general multiplication rule (for non-independent events)


26/52

Venn diagrams for

(a) event (notE

)(b) event (A &B)

(c) event (A orB)


27/52

Event (B & C)

1/13 X 1/4 = 1/52


28/52

Event (B orC)

16/52 = 4/52 + 13/52-1/52


29/52

Event (C&D)

3/52 = 3/13 X 1/4


30/52

Mutually Exclusive Events

Two or more events are said to bemutually exclusive if at

most one of them can occur when the experiment is performed,

that is, if no two of them have outcomes in common


31/52

Two mutually exclusive events


32/52

(a) Two mutually exclusive events

(b) Two non-mutually exclusive events


33/52

(a) Three mutually exclusive events (b) Three non-

mutually exclusive events (c) Three non-mutually

exclusive events


34/52

The Special Addition Rule

If event A and event B are mutually exclusive, then

More generally, if events A, B, C, are mutually exclusive, then

That is, for mutually exclusive events, the probability that at least one of

the events occurs is equal to the sum of the individual probabilities.

( ) ( ) ( )BPAPBAP +=or

( ) ( ) ( ) ( ) ......oror CPBPAPCBAP ++=


35/52

Non-mutually exclusive events


36/52

The General Addition Rule

IfA andB are any two events, then

P(A orB) = P(A) + P(B) P(A &B).

In words, for any two events, the

probability that one or the other occurs

equals the sum of the individualprobabilities less the probability that both

occur.


37/52

P(A or B): Spade or Face Card

P (spade) + P (face card) P (spade & face card) = 1/4 + 3/13 3/52

= 22/52


38/52

The Special Multiplication Rule (for independent events)

If eventsA, B, C, . . . are independent, then

P(A &B & C & ) = P(A) P(B) P(C).

What is the probability of all of these events occurring:

1. Flip a coin and get a head

2. Draw a card and get an ace

3. Throw a die and get a 1

P(A &B & C) =P(A) P(B) P(C) = 1/2 X 1/13 X 1/6


39/52

Conditional Probability: For non-independent events

The probability that eventB occurs given that eventA

has occurred is called aconditional probability. It is

denoted by the symbolP(B |A), which is read theprobability ofB givenA. We callA thegiven event.


40/52

Contingency Table for Joint Probabilities


41/52

Contingency table for age and rank of faculty members

(using frequencies)


42/52

The Conditional-Probability Rule

IfA andB are any two events, then

In words, for any two events, the conditional

probability that one event occurs given that the other

event has occurred equals the joint probability of the

two events divided by the probability of the givenevent.

.

)(

)&()|(

AP

BAPABP =

)&( BAP


43/52

.)(

)&()|(

AP

BAPABP =

The Conditional-

Probability Rule

P(R3 |A4 ) =

= 36/253

= 0.142

P(A4 |R3 ) =

= 36/320

= 0.112


44/52

Joint probability

distribution (using

proportions)

.)(

)&()|(

AP

BAPABP =

P(R3 |A4 ) =

= 0.031/0.217

= 0.142

P(A4 |R3 ) =

= 0.031/.0275

= 0.112


45/52

Contingency table of marital status and sex

(using proportions)


46/52

.

)(

)&()|(

AP

BAPABP =Joint probability

distribution (using

proportions)


47/52

The General Multiplication Rule

IfA andB are any two events, thenP(A &B) = P(A) P(B |A).

In words, for any two events, their joint probability

equals the probability that one of the events occurs times

the conditional probability of the other event given thatevent.

Note: Either

1) The events are independent and then

P(A &B) =P(A) P(B).

Or

2) The events are not independent and then a

contingency table must be used


48/52

Independent Events

EventB is said to be independent of eventA if theoccurrence of eventA does not affect the probability that

eventB occurs. In symbols,

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

This means that knowing whether eventA has occurredprovides no probabilistic information about the

occurrence of eventB.

Class Fr So Ju Se

Male 40 50 50 40 | 180

Female 80 100 100 80 | 360

120 150 150 120 | 540


49/52

Probability and the Normal Distribution

What is the probability of randomlyselecting an individual with an I.Q. between

95 and 115? Mean 100, S.D. 15.

Find thez-score for 95 and 115 andcompute the area between


50/52

More Preview of Experimental DesignUsing probability to evaluate a treatment effect. Values that are extremely

unlikely to be obtained from the original population are viewed as

evidence of a treatment effect.


51/52


52/52

Documents

06b Probability