Chapter 7
PROBABILITY AND
UNCERTAINTY
Probability distribution
2
Probability distribution
3
Possibility distribution
4
Comparison of Possibility and
Probability
5
Fuzzy Events
•
• Example
–
• The probability of such fuzzy events
– Crisp probability (Crisp value)
– Fuzzy probability (Fuzzy set)
6
Crisp Probability of Fuzzy Event
7
Crisp Probability of Fuzzy Event
• Example:
1
1
0
1
2
0
1
2
8
Crisp Probability of Fuzzy Event
9
Crisp Probability of Fuzzy Event
10
Crisp Probability of Fuzzy Event
11
Fuzzy Probability of Fuzzy Event
12
Fuzzy Probability of Fuzzy Event
13
Fuzzy Probability of Fuzzy Event
14
Fuzzy Probability of Fuzzy Event
1 0.8 0.5 0.3
15
Fuzzy Probability of Fuzzy Event
16
Fuzzy Probability of Fuzzy Event
17
Uncertainty
18
Measure of Fuzziness
.
19
Measure of Fuzziness
20
Measure of Fuzziness
21
Measure of Fuzziness
• Using entropy
• Using metric distance
22
Measure using Entropy
23
Measure using Entropy
24
Measure using Entropy
25
Measure using Entropy
26
Measure using Entropy
max
27
Measure using Entropy
28
Measure using Entropy
29
Measure using Entropy
30
Measure of the fuzziness using
metric distance
31
Measure of the fuzziness using
metric distance
32
Measure of the fuzziness using
metric distance
max
33
Measure of the fuzziness using
metric distance
Normalized :
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1
C: 0
0
1
C: 0
0
0
0.5
0.4
0.3
0.2
0.1
0.5
0.4
0.3
0.2
0.1
a b c
a b c
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Thanks for your attention!
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