topic 4 - probability (old notes)

7/30/2019 Topic 4 - Probability (Old Notes)

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TOPIC 4: PROBABILITY

Learning Objectives: Explain and illustrate the meaning of the

following probability concepts:

- mutually exclusive events- independent events

Apply the laws of addition and multiplication

to solve basic probability problems. Solve basic probability problems involving

conditional probability.

Apply tree techniques to solve probability


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PROBABILITY

Definitions

A probability is a measure of the likelihood that an event inthe future will happen, is denoted byP. It can only assume a

value between 0 and 1.

A value near zero means the event is not likely to happen.

A value near one means it is likely.

Two Properties of Probability:

1. The probability of an event always lies in the range 0 and

1. 0 P(A) 1

2. The sum of probabilities of all events for an experiment is

always 1.

2

1...)()()()( 321 EPEPEPEP i


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PROBABILITY

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Probability Examples


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PROBABILITYDefinitionscontinued

An experimentis the observation of someactivity or the act of taking some

measurement.

An outcome is the particular result of an

experiment.

An event is the collection of one or moreoutcomes of an experiment.

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PROBABILITYExamples:

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PROBABILITYClassical Probability Rule to Find Probability:

P(A) = Number of outcomes favorable to ATotal number of outcomes for the experiment

Example:

1. Find the probability of obtaining a head and the probabilityof obtaining a tail for one toss of a coin.

2. Find the probability of obtaining an even number in one roll

of a die.

3. In a group of 500 women, 80 have played golf at least once.

Suppose one of these 500 women is randomly selected.

What is the probability that she has played golf at least

once? 6


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PROBABILITYConditional Probability- is the probability that an event will

occur given that another event has already occurred. IfAandB are two events, then the conditional probability ofA

givenB is written as

P(A B)

and read as the probability ofA given thatB has already

occurred

Example: Two-Way Classification of Employee Responses

with Totals

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In Favor Against TotalMale 15 45 60

Female 4 36 40

Total 19 81 100


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PROBABILITY1. Compute the conditional probability

P(in favor male) for the data on 100 employeesgiven in the table above.

2. Calculate the conditional probability that a

randomly selected employee is a female given thatthis employee is in favor of paying high salaries to

CEOs.

SOLUTIONS:

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PROBABILITY Events are mutually exclusiveif the occurrence of any one

event means that none of the others can occur at the same

time.

Simply events that cannot occur together are said to be

mutually exclusive events.

Example:1. Consider the following events for one roll of a die:

A = an even number is observed = {2, 4, 6}

B = an odd number is observed = {1, 3, 5}C= a number less than 5 is observed = {1, 2, 3, 4}

Are eventsA andB mutually exclusive? Are eventsA and C

mutually exclusive?9


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PROBABILITY

Example cont

2. Consider the following two events for a randomlyselected adult:

Y= this adult has shopped on the internet at least once

N= this adult has never shopped on the internet.

Are events YandNmutually exclusive?

SOLUTIONS:

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PROBABILITY Events are independent if the occurrence of one

does not affect the probability of the occurrence ofthe other. In other words,A andB are independent

events if

either P(AB) =P(A) orP(BA) =P(B)

Example:

1. Refer to the information on 100 employees given inthe previous table above. Are Events female (F)

and in favor (A) independent?

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PROBABILITYExample:

2. A box contains a total of 100 CDs that weremanufactured on two machines. Of them, 60 were

manufactured on Machine 1. of the total CDs, 15 are

defective. Of the 60 CDs that were manufactured onMachine 1, 9 are defective. Let D be the event that a

randomly selected CD is defective, and let A be the

event that randomly selected CD was manufactured

on Machine 1. Are events D and A independent?

SOLUTION:

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PROBABILITYRules for Computing Probabilities

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1. Rules of Addition

Special Rule of Addition - If two eventsA andB are mutually exclusive, theprobability of one orthe other eventsoccurring equals the sum of their

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

The General Rule of Addition - If A and

B are two events that are not mutuallyexclusive, then P(A or B) is given bythe following formula:

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


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PROBABILITYExample:

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What is the probability that a card chosen at random

from a standard deck of cards will be either a king or

a heart?

P(A orB) = P(A) +P(B) -P(A andB)

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

= 16/52, or .3077


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PROBABILITYRules for Computing Probabilities

2. Special Rule of Multiplication

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The special rule of multiplication requires that

two eventsA andB are independent.

Two eventsA andB are independentif theoccurrence of one has no effect on the probability

of the occurrence of the other.

This rule is written: P(A andB) =P(A)P(B)


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PROBABILITYExample:

A survey by the American Automobile association(AAA) revealed 60 percent of its members made

airline reservations last year. Two members are

selected at random. What is the probability both

made airline reservations last year?

SOLUTION

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PROBABILITY


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PROBABILITYSolution:The probability the first member made an airline

reservation last year is .60, written asP(R1) =.60

The probability that the second member selected

made a reservation is also .60, so P(R2) = .60.Since the number of AAA members is very

large, you may assume that

R1 andR2 are independent.

P(R1 and R2) = P(R1)P(R2) = (.60)(.60) = .36

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PROBABILITYGeneral Multiplication Rule

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Thegeneral rule of multiplicationis used to find the jointprobability that two events will occur.

Use the general rule of multiplication to find the joint

probability of two events when the events are not

independent.

It states that for two events,A andB, the joint probability that

both events will happen is found by multiplying the

probability that eventA will happen by the conditional

probability of eventB occurring given thatA has occurred.

PROBABILITY


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PROBABILITYExample:

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A golfer has 12 golf shirts in his closet. Suppose 9 of

these shirts are white and the others blue. He gets

dressed in the dark, so he just grabs a shirt and puts

it on. He plays golf two days in a row and does not

do laundry.

What is the likelihood both shirts selected are white?

SOLUTION

PROBABILITY


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PROBABILITY

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SOLUTION:

The event that the first shirt selected is white is W1.The probability isP(W1) = 9/12

The event that the second shirt selected is also white

is identified as W2. The conditional probability thatthe second shirt selected is white, given that the first

shirt selected is also white, isP(W2 | W1) = 8/11.

To determine the probability of 2 white shirts being

selected we use formula: P(AB) = P(A) P(B|A)

P(W1 and W2) = P(W1)P(W2 |W1) = (9/12)(8/11) =

0.55


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PROBABILITYTree Diagrams

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A tree diagramis useful for portraying conditionaland joint probabilities. It is particularly useful for

analyzing business decisions involving several

stages.

A tree diagramis a graph that is helpful in

organizing calculations that involve several stages.

Each segment in the tree is one stage of the

problem. The branches of a tree diagram are

weighted by probabilities.

PROBABILITY


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PROBABILITYExample:

The probability that a person is in favor of genetic

engineering is 0.55 and that a person is against it is

0.45. the two persons are randomly selected, and it

is observed whether they favor or oppose genetic

engineering.

a.) Draw a tree diagram for this experiment.

b.) Find the probability that at least one of the two

persons favors genetic engineering.

SOLUTION

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topic 4 - probability (old notes)

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