Probability concepts-applications-1235015791722176-2

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Probability Probability Concepts and Concepts and ApplicationsApplications

IntroductionIntroduction

• Life is uncertain!

• We must deal with risk!

• A probability is a numerical statement about the likelihood that an event will occur

Basic Statements About Basic Statements About ProbabilityProbability

1. The probability, P, of any

event or state of nature

occurring is greater than or

equal to 0 and less than or

equal to 1.

That is: 0 P(event) 1

2. The sum of the simple

probabilities for all possible

outcomes of an activity must

equal 1.

ExampleExample

• Demand for white latex paint at

Diversey Paint and Supply has

always been 0, 1, 2, 3, or 4

gallons per day. (There are no

other possible outcomes; when

one outcome occurs, no other

can.) Over the past 200 days,

the frequencies of demand are

represented in the following

table:

Example - continuedExample - continued

Quantity

Demanded

(Gallons)

Number of Days

Total 200

Frequencies of DemandFrequencies of Demand

Example - continuedExample - continued

Quant. Freq.

Demand (days)

Total days = 200

Probability

(40/200) = 0.20

(80/200) = 0.40

(50/200) = 0.25

(20/200) = 0.10

(10/200) = 0.05Total Prob = 1.00

Probabilities of DemandProbabilities of Demand

Types of ProbabilityTypes of Probability

Objective probability:

Determined by experiment or observation:• Probability of heads on coin flip• Probably of spades on drawing card

from deck

occurrencesor outcomes ofnumber Total

occursevent timesofNumber )( eventP

Types of ProbabilityTypes of Probability

Subjective probability:

Based upon judgement

Determined by:

• judgement of expert

• opinion polls

• Delphi method

• etc.

Mutually Exclusive EventsMutually Exclusive Events

• Events are said to be mutually

exclusive if only one of the

events can occur on any one

Collectively Exhaustive Collectively Exhaustive EventsEvents

• Events are said to be

collectively exhaustive if the list

of outcomes includes every

possible outcome: heads and

tails as possible outcomes of

coin flip

ExampleExampleOutcome

of Roll

Probability

Total = 1

Rolling a die has six possible outcomes

ExampleExample

Outcome

of Roll = 5

Die 1 Die 2

Probability

Rolling two

dice results in

a total of five

showing.

There are a

total of 36

possible

outcomes.

Probability : Probability : Mutually Exclusive Mutually Exclusive

P(event A or event B) =

P(event A) + P(event B)

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

P(spade or club) = P(spade) + P(club)

= 13/52 + 13/52

= 26/52 = 1/2 = 50%

Probability:Probability: Not Mutually Exclusive Not Mutually Exclusive

P(event A or event B) =

P(event A) + P(event B) -

P(event A and event B both occurring)

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

P(A and B)P(A and B)(Venn Diagram)(Venn Diagram)

P(A) P(B)

P(A and B

P(A or B)P(A or B)

P(A) P(B) P(A and B)

P(A or B)

Statistical DependenceStatistical Dependence

• Events are either

• statistically independent (the

occurrence of one event has no

effect on the probability of

occurrence of the other) or

• statistically dependent (the

occurrence of one event gives

information about the occurrence

of the other)

Probabilities - Probabilities - Independent EventsIndependent Events

• Marginal probability: the probability of an event occurring:

[P(A)]• Joint probability: the probability of

multiple, independent events, occurring at the same time

P(AB) = P(A)*P(B)

• Conditional probability (for

independent events):

• the probability of event B given that

event A has occurred P(B|A) = P(B)

• or, the probability of event A given that

event B has occurred P(A|B) = P(A)

Probability(A|B) Probability(A|B) Independent EventsIndependent Events

P(A|B)P(B|A)

Statistically Statistically Independent EventsIndependent Events

1. P(black ball drawn on first draw)

• P(B) = 0.30 (marginal probability)

2. P(two green balls drawn)

• P(GG) = P(G)*P(G) = 0.70*0.70 = 0.49 (joint probability for two independent events)

A bucket contains 3 black balls, and 7 green balls. We draw a ball from the bucket, replace it, and draw a second ball

Statistically Independent Statistically Independent Events - continuedEvents - continued

1. P(black ball drawn on second

draw, first draw was green)

• P(B|G) = P(B) = 0.30

(conditional probability)

2. P(green ball drawn on second

draw, first draw was green)

• P(G|G) = 0.70

(conditional probability)

Probabilities - Probabilities - Dependent EventsDependent Events

• Marginal probability: probability of

an event occurring P(A)

• Conditional probability (for

dependent events):

• the probability of event B given that

event A has occurred P(B|A) =

P(AB)/P(A)

• the probability of event A given that

event B has occurred P(A|B) =

P(AB)/P(B)

Probability(A|B)Probability(A|B)

P(A|B) = P(AB)/P(B)

P(AB) P(B)P(A)

Probability(B|A)Probability(B|A)

P(B|A) = P(AB)/P(A)

P(AB)P(B) P(A)

Statistically Dependent Statistically Dependent EventsEvents

Assume that we have an urn containing 10 balls of the following descriptions:

•4 are white (W) and lettered (L)

•2 are white (W) and numbered N

•3 are yellow (Y) and lettered (L)

•1 is yellow (Y) and numbered (N)

• P(WL) = 4/10 = 0.40

• P(WN) = 2/10 = 0.20

• P(W) = 6/10 = 0.60

• P(YL) = 3/10 = 0.3

• P(YN) = 1/10 = 0.1

• P(Y) = 4/10 = 0.4

Statistically Dependent Statistically Dependent Events - ContinuedEvents - Continued

Then:• P(L|Y) = P(YL)/P(Y)

= 0.3/0.4 = 0.75

• P(Y|L) = P(YL)/P(L)

= 0.3/0.7 = 0.43

• P(W|L) = P(WL)/P(L)

= 0.4/0.7 = 0.57

Joint Probabilities, Joint Probabilities, Dependent EventsDependent Events

Your stockbroker informs you that if

the stock market reaches the 10,500

point level by January, there is a

70% probability the Tubeless

Electronics will go up in value.

Your own feeling is that there is

only a 40% chance of the market

reaching 10,500 by January.

What is the probability that both the

stock market will reach 10,500

points, and the price of Tubeless

will go up in value?

Joint Probabilities, Dependent Joint Probabilities, Dependent Events - continuedEvents - continued

P(MT) =P(T|M)P(M)

= (0.70)

(0.40)

= 0.28

Let M represent

the event of the

stock market

reaching the

10,500 point

level, and T

represent the

event that

Tubeless goes

Revising Probabilities: Revising Probabilities: Bayes’ TheoremBayes’ Theorem

Bayes’ theorem can be used to calculate revised or posterior probabilities

Prior Probabilities

Bayes’ Process

Posterior Probabilities

New Information

General Form of General Form of Bayes’ TheoremBayes’ Theorem

Aevent theof complementA where

)()|()()|(

)()|()|(

APABPAPABP

APABPBAP

ABPBAP

die.unfair"" is Aevent then the

die,fair"" isA event theifexample,For

Posterior ProbabilitiesPosterior ProbabilitiesA cup contains two dice identical in

appearance. One, however, is fair (unbiased), the other is loaded (biased). The probability of rolling a 3 on the fair die is 1/6 or 0.166. The probability of tossing the same number on the loaded die is 0.60.

We have no idea which die is which, but we select one by chance, and toss it. The result is a 3.

What is the probability that the die rolled was fair?

Posterior Probabilities Posterior Probabilities ContinuedContinued

• We know that:P(fair) = 0.50 P(loaded) = 0.50

• And:

P(3|fair) = 0.166 P(3|loaded) = 0.60

• Then:P(3 and fair) = P(3|fair)P(fair)

= (0.166)(0.50)

= 0.083

P(3 and loaded) = P(3|loaded)P(loaded)

= (0.60)(0.50)

= 0.300

Posterior Probabilities Posterior Probabilities ContinuedContinued

• A 3 can occur in combination with

the state “fair die” or in combination

with the state ”loaded die.” The sum

of their probabilities gives the

unconditional or marginal probability

of a 3 on a toss:

P(3) = 0.083 + 0.0300 = 0.383.

• Then, the probability that the die

rolled was the fair one is given by:

0.22 0.383

3) andP(Fair 3)|P(Fair

Further Probability Further Probability RevisionsRevisions

• To obtain further information as

to whether the die just rolled is

fair or loaded, let’s roll it again.

• Again we get a 3.

Given that we have now rolled

two 3s, what is the probability

that the die rolled is fair?

Further Probability Further Probability Revisions - continuedRevisions - continued

P(fair) = 0.50, P(loaded) = 0.50 as before

P(3,3|fair) = (0.166)(0.166) = 0.027

P(3,3|loaded) = (0.60)(0.60) = 0.36

P(3,3 and fair) = P(3,3|fair)P(fair)

= (0.027)(0.05)

= 0.013

P(3,3 and loaded) = P(3,3|loaded)P(loaded)

= (0.36)(0.5)

= 0.18

P(3,3) = 0.013 + 0.18 = 0.193

933.00.193

P(3,3)

Loaded) and P(3,3 3,3)|P(Loaded

067.00.193

P(3,3)

Fair) and P(3,33,3)|P(Fair

To give the final comparison:

P(fair|3) = 0.22

P(loaded|3) = 0.78

P(fair|3,3) = 0.067

P(loaded|3,3) = 0.933

Probability concepts-applications-1235015791722176-2

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