Chapter 1 Probability

5/28/2018 Chapter 1 Probability

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Chapter 1

Basic Probability(Week 1)

L1Introduction to Basic Probability -

Sample Spaces and Events


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Learning Objectives:

At the end of the lecture student should be able to:

- Define and construct sample space of an experiment.

- Define random events, identify types of

events, apply Venn Diagram and laws to find event setincluding intersection, union and complement.

- Identify mutually exclusive and exhaustive events.

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Probability refers to the study of randomnessand uncertainty.

Probability forms the basis with which we can make inferences

about a population based on the distribution and it provides

methods for quantifying the chances or likelihood associated with

various outcomes. Probability helps to explain a lot of everyday

occurrences and we actually discuss it frequently.

We also use it everyday in engineering. Example: the probability of

a good part being produce, the reliability of a new machine

(reliabilities are actually probabilities) etc.

An engineer wants to be fairly certain that the percentage of good

rods is at least 90%; otherwise he will shut down the process for

recalibration. How certain can be that at least 90% of the 1000

rods are good?

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Definitions

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Random Process: any process whose possible results are known but actualresults cannot be predicted with certainty in advance.

Outcome : each possible result for a random process

Experiment: process by which an observation or measurement is obtained(yield outcomes)

How can we model random experiment?

Sample Space: denoted S, is the set of all possible outcomes of an experiment.

Eventis any collection (subset) of outcomes contained in the sample space S.

An event is simpleif it consists of exactly one outcome andcalled compoundevent if it consists of more than one outcome.

Null event: An event with no outcomes (= impossible event, empty set).


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Sample Spaces and Events

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Example1.

Roll a die Sample space:

S = {1, 2, 3, 4, 5, 6}

Simple events(or outcomes):

E1: observe a 1= {1} E3 = {3} E4 = {4}

E2 = {2} E5 = {5} E6 = {6}

Compound events:A : observe an odd number = {1, 3, 5}B : observe a number greater than or equal to 4 = {4, 5, 6}


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Example 2

Toss a coin three times and note the number of heads

S = {0, 1, 2, 3 }

The lifetime of a machine (in days)

S = {t | t 0 }= [0, )

The working state of a machine

S = {working, fail }

The number of calls arriving at a telephone exchange during aspecific time interval

S = {0, 1, }


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Example 3: Each message in a digital communication system is classified as to

whether it is received within the time specified by the system design. If

3 messages are classified, what is an appropriate sample space for

this experiment?

To generate the sample space, we can use a tree diagram

n

y

y

y

y

yy

yn

n

n

n

n

n

Message 1

Message 2Message 3

S = { yyy, yyn, yny, ynn,

nyy, nyn, nny, nnn}


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More Definitions

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The unionof events A and B, denoted by A U B and read A orB is the eventconsisting of all outcomes that are either in A orin B or in both events.

The intersectionof A and B, denoted by A B and read A andB, is the event

consisting of all outcomes that are in both A and B.

The complementof event A, A, is the event of all outcomes in the sample

space S that are not contained in event A.

If two events A and B have no outcomes in common they are said to be

mutually exclusiveor disjoint events. This means if one of the event occurs

the other cannot.


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Venn Diagram

Graphical display of events in a sample space.

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Example 4

A digital scale is used that provide weights to the nearest

gram.Let event A: a weight exceeds 11 grams

B: a weight is less than or equal to 15 grams

C: a weight is greater than or equal to 8 grams andless than 12 grams.

a) What is the sample space for this experiment?

Describe the following events

b)A UB c) A B

d)A e)A UB UA UC

f) (A UC) g) A B C

h) B C i) A U (B C)


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S = nonnegative integers from 0 to the largest integer that can

be displayed by the scale.S = {0, 1, 2, 3, }

Let X represent weight.

A = the event that X > 11 or {12,13,14,..}

B = the event that X 15 or = {0,1, 2, 3, .....15}

C = the event that 8 X


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f) AC = {8, 9 ,10, 11, 12,,13, } or { X: X 8}

Thus (A C) = {0, 1, 2, , 7}or { X: X < 8 }

g) A

B

C = {A

B} C

={12, 13,14, 15} {8, 9, 10, 11}=

A ={12,13,14,..}B ={0,1, 2, 3, .....15}

C ={8, 9, 10, 11}


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h) B= { X: X > 15}.

Therefore, BC would be the empty set.They have no outcomes in common or

i) B C = { X: 8 X

Chapter 1 Probability

Documents

sample space s

probability ofa good

die sample space

events5282018 chapter

construct sample space

possible outcomes

random experiment

random events