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Lecture 15:
Probability review
Bayes Nets intro
Prof. Julia Hockenmaier
juliahmr@illinois.edu
http://cs.illinois.edu/fa11/cs440
CS440/ECE448: Intro to ArtiFcial Intelligence
What is the probability of…?
2
P(
)
= 2/15
P(
blue
) = 5/15
P(
blue

) = 2/5
P(
)
= 1/15
P(
red
)
= 5/15
P(
)
= 5/15
P(
or
) = 2/15
P(

red
) = 3/5
Some terminology…
Trial:
e.g. picking a shape
Sample space
!
:
the set of all possible
outcomes (e.g. all kinds of shapes)
Event
"
˧
!
:
an actual outcome of a trial
(a subset of
!
)
3
CS440/ECE448: Intro AI
Coin tossing
Bernoulli distribution:
Probability of success (
head
) in single yes/no trial
The probability of
head
is
p
.
The probability of
tail
is
1
−
p
.
Binomial distribution:
Prob. of getting
k
heads in
n
independent
yes/no trials
4
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View Full DocumentRolling a die
Categorical distribution:
Prob. of getting one of
K
outcomes in a single trial
The probability of outcome c
i
is
p
i
(
!
p
i
= 1
)
Multinomial distribution:
Prob. of observing each possible outcome c
i
exactly
x
i
times in a sequence of
n
yes/no trials
5
Laws of probability
Ω
B
6
CS440/ECE448: Intro AI
A
A
ˬ
B
P(
!
) = 1
˲
A
⊆
!
:
0
≤
P(A)
≤
1
˲
A,B
⊆
!
: P(A
∩
B)
≤
P(A)
˲
A,B
⊆
!
: P(A
∪
B) = P(A) + P(B) – P(A
∩
B)
Random variables
A function which maps every element in the
sample space to some value.
Boolean random variables:
heads or tails?
Categorical random variables:
color, shape
Continuous random variables:
size, height,…
7
¬A
Discrete random variables
The possible outcomes of discrete random
variables (=atomic events)
partition the
sample space
8
CS440/ECE448: Intro AI
A
red
yellow
blue
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 Spring '08
 Levinson,S

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