هذا العرض يشرح أساسيات التفكير الاحتمالي (Probabilistic Reasoning) وكيفية استخدامه في الذكاء الاصطناعي وحل المشكلات.

قوانين الاحتمالات و الاحصاء

1. October 2025 [email protected]
PROBABILISTIC
REASONING

2. 1
Sample Space
The roll of a dice
The toss of (flipping) a coin
Random (Statistical) Experiment:
•An experiment <with known outcomes> whose outcome
cannot be predicted with certainty, before the experiment
is run.

3. 2
Sample Space
Sample Space (?????? ):•
•
Set of ALLpossible outcomes of a random experiment.

countable infinite set of outcomes.
• A sample space is continuousif it contains an interval
(either finite or infinite) of real numbers.
A sample space is discrete if it consists of a finite or

4. 3
Sample Space
Sample Space (?????? ):
?????? = {1,2,3,4,5,6}
• Set of ALLpossible outcomes of a random experiment.
The roll of a dice
Discrete
Each outcome in a sample space is called
an element or a member of the sample
space, or simply a sample point.

5. 4
Sample Space
Sample Space (?????? ):
• Set of ALLpossible outcomes of a random experiment.
Flipping a coin
&#3627408454; = {??????&#3627408466;??????&#3627408465;, &#3627408455;??????????????????}
&#3627408454; = {??????, &#3627408455;}

6. Tree Diagrams:
Sample spaces can also be described graphically with
diagrams.
tree
&#3627408454; = {????????????,??????&#3627408455;,&#3627408455;??????,&#3627408455;&#3627408455;}
5
Sample Space

7. 6
Events
Event ( ):
??????
•A result of none, one, or moreoutcomes in the sample
space. An event is a subset of the sample space of a
random experiment.
The roll of a dice
?????? = {1,2,3,4,5,6}
?????? = {2,4,6}
Even Numbers

8. Venn Diagrams:
Diagrams are often used to portray relationships between
sets, and these diagrams are also used to describe
relationships between events. We can use Venn diagrams
to represent a sample space and events in a sample space.
7
Events
?????? ??????

9. Example1:
??????= 1,2,3,4,5,6,7
&#3627408436;= 1,2,4,7
&#3627408437;= 1,2,3,6
&#3627408438;= 1,3,4,5
8
Events

10. Example2:
35
Events
9

11. Example3:
10
Events

12. Example4:
11
Events

13. Example5:
12
Events

14. Example6:
??????= 1,2,3,4,5,6,7
&#3627408436;= 1,2,4,5,7
&#3627408437; = 1,2
&#3627408438; = 4,6
13
Events
1
5
2
7
4
6

15. Several Results:
14
Events

16. 15
Counting Techniques
Multiplication (Product) Rule: Suppose that a procedure
can be broken down into a sequenceof twotasks. If there
are ??????&#3627409359;ways to do the first
task and for each of these ways of doing the first task, there
are ??????&#3627409360;ways to do the second task, then there are ??????&#3627409359;??????&#3627409360;
ways to do the procedure and so forth.
The total number of ways to complete the operation is
??????1×??????2×⋅⋅⋅×????????????

17. 16
Counting Techniques
Permutations (1/3):
Another useful calculation finds the number of ordered
sequences of the elements of a set. Consider a set of
.
A permutation of the elements is an ordered sequence of the
elements. For example, &#3627408462;&#3627408463;&#3627408464;, &#3627408462;&#3627408464;&#3627408463;, &#3627408463;&#3627408462;&#3627408464;, &#3627408463;&#3627408464;&#3627408462;, &#3627408464;&#3627408462;&#3627408463;, and &#3627408464;&#3627408463;&#3627408462;are
all of the permutations of the elements of ??????.
elements, such as
?????? = {&#3627408462;, &#3627408463;, &#3627408464;}
3×2×1=6

18. 17
Counting Techniques
Permutations (2/3):
Ex.
and

The number of permutations of the four letters
&#3627408465;
.
, , ,
&#3627408462;&#3627408463;&#3627408464;
will be
4! = 24

19. 18
Counting Techniques
Permutations (3/3):
In some situations, we are interested in the number of
arrangements of only some of the elements of a set.
The number of permutations of subsets of ??????elements
selected from a set of ??????different elements is

20. 19
Counting Techniques
Permutations of Similar Objects:

21. 20
Counting Techniques
Combinations (1/2):
Another counting problem of interest is the number of
subsets of ??????elements that can be selected from a set of ??????
elements. Here, order is not important. These are called
combinations.

22. 21
Counting Techniques
Combinations (2/2):
The number of combinations, subsets of ??????elements that
??????
??????
can be selected from a set of ??????elements, is denoted as
or ?????????????????? and

23. 22
Probability of an Event
Equally Likely Outcomes:
•
•
?????? ??????
Is a sample space, is an event
??????⊆??????
1≥?????? ?????? ≥0
????????????=
#of outcomes in event(??????)
Total # of outcomes in sample space(??????)
??????
??????
??????
=
??????

24. 23
Probability of an Event
Axioms of Probability:
??????is a sample space,
??????⊆??????
????????????=&#3627409359;
??????∅=&#3627409358;
&#3627409358;≤????????????≤&#3627409359;
????????????′=&#3627409359;−??????(??????)
??????is an event

25. FOR YOUR ATTENTION THANK YOU