A Short Review of Probability Theory

7/31/2019 A Short Review of Probability Theory

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Bustamin Bin AhmadMuhammad Norhadri Bin Mohd Helmi

Tan Cheng Peng

Wong Wei Chieh



2-0 Introduction2-1 Terminologies

2-2 Axioms of Probability

2-3 Mutually Exclusive Events

2-4 Independent Events

2-5 Addition Rules

2-6 Conditional Probability

2-7 Multiplication and Total Probability Rules

2-8 Random Variables

2-9 Probability Distribution2-10 Probability vs Statistic

2-10 Probability Tools



5

Probability shows you thelikelihood, or chances, for eachof the various future outcomes,based on a set of assumptions

about how the world works.

2-1 Sample Spaces

and Events

2-2 Interpretations of

Probability

2-3 Addition Rules

2-4 Conditional

Probability

2-5 Multiplication and

Total Probability

Rules

2-6 Independence


2-8 Probability

Distribution

2-9 Statistic and

Probability




6

What is the probability that aflipped coin shows head up?

2-1 Sample Spaces

and Events


Probability

2-3 Addition Rules

2-4 Conditional

Probability


Total Probability

Rules

2-6 Independence


2-8 Probability

Distribution

2-9 Statistic and

Probability




8

2-0 Introduction


Probability

2-3 Addition Rules

2-4 Conditional

Probability

2-5 Multiplication andTotal Probability

Rules

2-6 Independence


2-8 Probability

Distribution

2-9 Statistic and

Probability


Sample Space, S

Outcome

Simple

event

Event, E Intersection,



9

2-0 Introduction


Probability

2-3 Addition Rules

2-4 Conditional

Probability

2-5 Multiplication andTotal Probability

Rules

2-6 Independence


2-8 Probability

Distribution

2-9 Statistic and

Probability


Sample space : the collection of all possible outcomesof a random circumstance

A simple event is one outcome in the sample space.

An event is a collection of one or more simple events(outcomes) in the sample space.Intersection

Complementary Events



Between 0 and 1

The sum of the probabilities over allpossible simple events is 1

2-0 Introduction

2-1 Sample Spaces

and Events

2-3 Addition Rules

2-4 Conditional

Probability


Total Probability

Rules

2-6 Independence


2-8 Probability

Distribution

2-9 Statistic and

Probability


Example:



Two events are mutually exclusive if they donot contain any of the same outcomes



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

Events, E1 = 'observe an odd number' = {1,3,5}

Events, E2 = 'observe an even number' = {2,4,6}



Two events are independent if theprobability that one event occursstays the same, no matter whetheror not the other event occurs.

P(A|B) = P(A)

2-1 Sample Spaces

and Events


Probability

2-3 Addition Rules

2-4 Conditional

Probability


Total Probability

Rules

2-6 Independence


2-8 Probability

Distribution

2-9 Statistic and

Probability




When?

What?2 events are independent if any of the following is

true:

P(A|B) = P(A)

P(B|A) = P(B) P(A∩B) = P(A)P(B)

- aka. Multiplication rule for independent event



Probability of first throw of dice that gives a „6‟ =1/6

Probability of second throw of dice that gives a „6‟= 1/6

Let A represent getting „6‟ on first throw.

Let B represent getting „6‟ on second throw.

Find probability of getting on both throws:

P(A∩B) = P(A)P(B) = (1/6)(1/6) = 0.36



Coin flip example:

One flip: Heads or Tails are mutuallyexclusive events

Two flips: The outcome of each flip isindependent.



= P(A) + P(B) - P(A ∩ B)



A + B - A∩B

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

A A∩B B



Probability of B given A, P(B|A)

P(B|A) =

Diagram

P(B|A) = P(A∩B)/P(A)

A A∩B

A∩B

A



A math teacher gave her class two tests. 25%of the class passed both tests and 42% of theclass passed the first test. What percent of those who passed the first test also passed

the second test?Solution:P(Second|First) = P(First and Second) = 0.25 = 0.60 = 60%

P(First) 0.42



From ,P(B|A) = P(A∩B)/P(A)

Rearrange the equation,

P(A∩B) = P(B|A)P(A) ………….. (i)

Interchange A and B,P(A∩B) = P(A|B)P(B) ………….. (ii)

Combining (i) and (ii), we called it the

P(A∩B) = P(B|A)P(A) = P(A|B)P(B)



Probability of first children is boy = 0.6;

P(A) = 0.6

Given that the first children is boy, the probabilityof second children is boy = 0.7;

P(B|A) = 0.7

What is the probability of ?Using Multiplication Rule:

=> P (A ∩ B) = P(B|A)P(A) = (0.6)(0.7) = 0.42



A A‟

Sample S:

For any events A and B:P(B) = P(B∩A) + P(B∩A‟) = P(B|A)P(A) +

P(B|A‟)P(A‟)



A A‟

BSampleS:

For any events A and B:P(B) = P(B∩A) + P(B∩A‟) = P(B|A)P(A) +

P(B|A‟)P(A‟)



A A‟

B B∩A B∩A‟

SampleS:

For any events A and B:P(B) = P(B∩A)+P(B∩A‟)

= P(B|A)P(A) + P(B|A‟)P(A‟)–



F = Product RejectedC = Product Contaminated

P(F|C) = 0.1 ; P(F|C‟)=0.005

P(C) = 0.2 ; P(C‟) = 0.8

Using Total Probability Rule:

P(F) = P(F|C)P(C) + P(F|C‟)P(C‟)= 0.1(0.2) + 0.005(0.8)

= 0.024





Not same with traditional way◦ e.g: Algebra

x + 3 = 7



Uncertain value that changes


















, ,



= number of student that will come to classtomorrow

= 0, 1, 2, 3, …, 10



Discrete random variable

Continuous random variable



Discrete random variable

Finite (countable) values



Continuous random variable

Infinite values

Need to be measured



Data types

Numerical Qualitative



=

()

Probability distribution function probability distributionRandom variable



=

()

Probability mass function

Probability density functionCumulative distribution function

Discrete probability distribution

Continuous probability distribution

Discrete

continuous



The normal distribution

The uniform distribution

The exponential distribution



The binomial distribution- Involves finite number of possibilities

The poisson distribution- Where the number count has no limit



Random Variable – a function or rule thatassigns a number to each outcome of anexperiment

RANDOM

VARIABLE

DISCRETE

RANDOM

VARIABLE

Probability Mass

Function

Binomial

Poisson

CONTINUOUS

RANDOM

VARIABLE

Probability

Density Function

Normal

Exponential



Random variables that take on afinite (or countable) number of

values.

•Sum of two dice (2,3,4,…,12) •Number of children (0,1,2,…) •Number in attendance at the

movies•Number of hired employees•Number of students coming toclass

Random variables that take onvalues in a continuum or

infinitely many values.

•Height•Weight•Time•

Time you can hold your breath•Lifetime of your cell phonebattery



68%

95%

99%

x



σ)

away from the mean (μ).

σ)




σ)




• The Standard normal distribution has

a mean of and a standard

deviation of .Parameters

• Any normal distribution can be

converted into a standard normal

distribution by getting a .Conversion





.







59

observed data to generalizationsabout how the world works.

2-1 Sample Spacesand Events


Probability

2-3 Addition Rules

2-4 Conditional

Probability


Total Probability

Rules

2-6 Independence


2-8 Probability

Distribution

2-9 Statistic and

Probability


Example:

-the seven hottestyears on record occurredin the most recent decade

- (perhapswithout justification)



60

An assumption about how the worldworks, figure out what kinds of datayou are likely to see.

2-1 Sample Spacesand Events


Probability

2-3 Addition Rules

2-4 Conditional

Probability


Total Probability

Rules

2-6 Independence


2-8 Probability

Distribution

2-9 Statistic and

Probability


Example:

- there is no globalwarming and ask how likelywe would be to get suchhigh temperatures as wehave been observingrecently

- probabilityprovides the justification forstatistics





Books: [1] „Applied Statistics and Probability for Engineers‟ by

Douglas Montgomery [2] „Probability and Statistics‟ by Moris H.DeGroot [3] „Handbook of Probability‟ by Rudas

Online Reference: [1] MIT Open Courseware

“Introduction to Probability and Statistics” http://ocw.mit.edu/courses/mathematics/18-05-introduction-to-probability-and-statistics-spring-

2005/lecture-notes/ http://www.cs.sunysb.edu/~skiena/jaialai/excerpts/node1

2.html Wikipedia

http://ocw.mit.edu/courses/mathematics/18-05-introduction-to-probability-and-statistics-spring-2005/lecture-notes/



http://www.cs.sunysb.edu/~skiena/jaialai/excerpts/node12.html























A Short Review of Probability Theory

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