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Probability Review
Probability Review-1
Probability Theory
• Mathematical description of relationships or occurrences
that cannot be predicted precisely
• An experiment is an activity whose outcome is subject to
random (i.e. chance or unknown) variation.
Examples:
– Flip a coin
– Toss a die
– …
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Sample Space
• The set of all possible outcomes of an experiment is
known as its sample space, denoted as S
Examples:
– Coin flipping
S={
}
– Tossing a die
S={
}
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Events
• An event is a collection of outcomes from a sample
space, denoted as E
Examples:
– Coin flipping
Event = Get a tail
– Tossing a die
Event = Get an even number : outcome = {2,4,6}
• An event E is said to occur if one of the outcomes with
which it is associated is realized during a replication of
the experiment
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Events
• Two events, E and F, are said to be mutually exclusive if
E∩F=
which means…
• The complement of event E, denoted Ec, is that unique
set such that E U Ec = S and E ∩ Ec = 
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Random Variable
• A random variable is a function that maps the sample space to the
real line.
Examples:
– Coin flipping
X = 1 if heads
0 if tails
– Tossing a die
W = (the number that shows on die)
• A random variable is discrete if the possible values it can assume
can be counted
• A random variable is continuous if it can assume any value in a
continuous subset of the real line
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Probability
• The probability associated with a particular event E, denoted P(E),
can be thought of as representing the relative likelihood of that event
occurring
• We will be generally thinking in terms of the probability of a random
variable taking a specific value
Examples:
– Coin flipping
P(X=1) =
– Tossing a die
P(W=6) =
P(W=7) =
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Axioms of Probability
• 0 ≤ P(E) ≤ 1 for any E
• P(S) = 1
• If {Ei, i=1,…,k} are mutually exclusive events,
then
 k  k
P  Ei    P( Ei )
 i 1  i 1
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Probability Distributions
• Describes probabilities of values a random variable could take
• Discrete
Examples:
– Coin flipping
P(X=x) =
½
0
– Tossing a die
P(W=w) = 1/6
0
1
0.8
0.6
0.4
0.2
0
if x={0,1}
otherwise
1
2
0.2
if w={1,2,3,4,5,6}
otherwise
0.15
0.1
0.05
0
1
2
3
4
5
6
• Continuous
Examples:
0
40
00
80
00
12
00
16 0
00
20 0
00
24 0
00
28 0
00
32 0
00
0
– Altitude of an airplane
Area under curve =
Probability Review-9
Common Probability Distributions
• Discrete
–
–
–
–
–
Discrete uniform
Poisson
Geometric
Binomial
…
• Continuous
–
–
–
–
–
–
–
Uniform
Exponential
Normal
Gamma
Beta
Triangular
…
Probability Review-10
PDF(PMF) vs. CDF
• Probability density
function (p.d.f.) denote f
• Probability mass function
(p.m.f.) denote f
f(x)=P(X=x)
• Cumulative distribution
function (c.d.f.) denote F
F(x)=P(X≤x)
Probability Review-11
Mean and Variance
• Mean (Expected Value)
E[X] =  =
 xf (x)
or

 xf ( x )dx

• Variance (Expected square distance from mean)
Var(X) = 2 = E[(X-E[X])2] = E[X2] – E[X]2
• Standard deviation (Spread)
  Var( X)
Probability Review-12
Examples of Mean and Variance
• Coin flipping
– Expected value =
– Variance =
– Standard deviation =
• Tossing a die
– Expected value =
– Variance =
– Standard deviation =
• Continuous uniform between 0 and 2
– Expected value =
– Variance =
– Standard deviation =
Probability Review-13
Conditional Probabilities
• Consider two experiments with S1={E1,…,Em} and
S2={F1,…,Fn}
• P(E|F) = P{experiment 1 gets outcome E given that
experiment 2 gets outcome F}
P( E  F )
P( E | F ) 
P( F )
• Example:
P(Ice cream sales > 10 cones | temperature = 85 F)
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Example 1 of Conditional Probabilities
• The king comes from a family of 2 children. What is the
probability that the other child is his sister?
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Example 2 of Conditional Probabilities
•
52% of the students at a certain college are females.
5% of the students in this college are majoring in
computer science. 2% of the students are women
majoring in computer science. If a student is selected
at random, find the conditional probability that
1) this student is female, given that the student is majoring
in computer science;
2) this student is majoring in computer science, given that
the student is female.
Probability Review-16
Bayes’ Theorem
P(   k | X  x j ) 
=
P(X  x j |    k ) P(   k )
P(X  x j )
P(X  x j |    k ) P(   k )
P(X  x


j
|   i ) P(  i )
i
Probability Review-17
Example 1 of Bayes’ Theorem
• Suppose that an insurance company classifies people
into one of three classes – good risks, average risks, and
bad risks. Their records indicate that the probabilities
that good, average, and bad risk persons will be involved
in an accident over a 1-year span are, respectively, 0.05,
0.15, and 0.30. If 20% of the population are “good risks”,
50% are “average risks”, and 30% are “bad risks”, what
proportion of people have accidents in a fixed year? If
policy holder A had no accidents in 1987, what is the
probability that he or she is a good risk?
Probability Review-18
Example 2 of Bayes’ Theorem
• Suppose that there was a cancer diagnostic test that
was 95% accurate both on those that do and those that
do not have the disease. If 0.4% of the population have
a cancer, compute the probability that a tested person
has cancer, given that his or her test result indicates so.
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