a short review of probability theory
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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
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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
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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
2-7 Random Variables
2-8 Probability
Distribution
2-9 Statistic and
Probability
2-10 Probability Tools
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What is the probability that aflipped coin shows head up?
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
2-7 Random Variables
2-8 Probability
Distribution
2-9 Statistic and
Probability
2-10 Probability Tools
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2-0 Introduction
2-2 Interpretations of
Probability
2-3 Addition Rules
2-4 Conditional
Probability
2-5 Multiplication andTotal Probability
Rules
2-6 Independence
2-7 Random Variables
2-8 Probability
Distribution
2-9 Statistic and
Probability
2-10 Probability Tools
Sample Space, S
Outcome
Simple
event
Event, E Intersection,
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2-0 Introduction
2-2 Interpretations of
Probability
2-3 Addition Rules
2-4 Conditional
Probability
2-5 Multiplication andTotal Probability
Rules
2-6 Independence
2-7 Random Variables
2-8 Probability
Distribution
2-9 Statistic and
Probability
2-10 Probability Tools
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
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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
2-5 Multiplication and
Total Probability
Rules
2-6 Independence
2-7 Random Variables
2-8 Probability
Distribution
2-9 Statistic and
Probability
2-10 Probability Tools
Example:
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Two events are mutually exclusive if they donot contain any of the same outcomes
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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}
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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
2-2 Interpretations of
Probability
2-3 Addition Rules
2-4 Conditional
Probability
2-5 Multiplication and
Total Probability
Rules
2-6 Independence
2-7 Random Variables
2-8 Probability
Distribution
2-9 Statistic and
Probability
2-10 Probability Tools
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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
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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
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Coin flip example:
One flip: Heads or Tails are mutuallyexclusive events
Two flips: The outcome of each flip isindependent.
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= P(A) + P(B) - P(A ∩ B)
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A + B - A∩B
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
A A∩B B
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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
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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
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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)
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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
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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‟)
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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‟)
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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‟)–
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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
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Not same with traditional way◦ e.g: Algebra
x + 3 = 7
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Uncertain value that changes
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Uncertain value that changes
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Uncertain value that changes
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Uncertain value that changes
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Uncertain value that changes
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Uncertain value that changes
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, ,
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= number of student that will come to classtomorrow
= 0, 1, 2, 3, …, 10
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Discrete random variable
Continuous random variable
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Discrete random variable
Finite (countable) values
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Continuous random variable
Infinite values
Need to be measured
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Data types
Numerical Qualitative
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=
()
Probability distribution function probability distributionRandom variable
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=
()
Probability mass function
Probability density functionCumulative distribution function
Discrete probability distribution
Continuous probability distribution
Discrete
continuous
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The normal distribution
The uniform distribution
The exponential distribution
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The binomial distribution- Involves finite number of possibilities
The poisson distribution- Where the number count has no limit
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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
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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
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68%
95%
99%
x
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σ)
away from the mean (μ).
σ)
away from the mean (μ).
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σ)
away from the mean (μ).
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• 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
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.
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observed data to generalizationsabout how the world works.
2-1 Sample Spacesand Events
2-2 Interpretations of
Probability
2-3 Addition Rules
2-4 Conditional
Probability
2-5 Multiplication and
Total Probability
Rules
2-6 Independence
2-7 Random Variables
2-8 Probability
Distribution
2-9 Statistic and
Probability
2-10 Probability Tools
Example:
-the seven hottestyears on record occurredin the most recent decade
- (perhapswithout justification)
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An assumption about how the worldworks, figure out what kinds of datayou are likely to see.
2-1 Sample Spacesand Events
2-2 Interpretations of
Probability
2-3 Addition Rules
2-4 Conditional
Probability
2-5 Multiplication and
Total Probability
Rules
2-6 Independence
2-7 Random Variables
2-8 Probability
Distribution
2-9 Statistic and
Probability
2-10 Probability Tools
Example:
- there is no globalwarming and ask how likelywe would be to get suchhigh temperatures as wehave been observingrecently
- probabilityprovides the justification forstatistics
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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