Download Introduction to probability (1)

Introduction to probability (1)
Introduction to probability (1)
• Probability is the language we use
to model uncertainty, of the
outcomes such as a scientific
experiments or any action.
• These outcomes could have been
different.
• Probability
is
underlying
foundation on which the important
methods of inferential statistics are
built.
Introduction to probability (1)
• Example: Suppose that you were to win
the top prize is a lotto five times, there
would
be
investigations
and
accusations that you were some how
cheating.
• This is incredibly; this is exactly how
statisticians think.
• We reject luck as a reasonable
explanation based on very low
probabilities, so the statisticians’ use
the rare event rule.
Introduction to probability (1)
• Rare Event Rule for Inferential
Statistics
• If under a given assumption (such as a
lottery being fair), the probability of a
particular observed even (such as five
lottery wins) is exactly small, we
conclude that the assumption is
probably not correct.
• In any experiment it produces
outcomes and we should no some
definitions to those outcomes:
Introduction to probability (1)
• Definitions:
• An Event: It is any collection of results or
outcomes of procedures.
• Simple Event: It is an outcome or an event
that can’t be further broken down into
simpler components.
• Sample Space: The sample space for a
procedure consists of all possible simple
events. That is, the sample space consists of
all outcomes that cannot be broken down
any further.
• Sample space is represented by the symbol S
Introduction to probability (1)
• Example:
Procedure
Example of event Sample space S
Roll one die
5,4 [are simple event] {1, 2, 3, 4, 5, 6}
Roll two dies 7(not a simple event)
{1-1,1-2,…..,6-6}
Introduction to probability (1)
•
Before we start in probability we
should know about:
1. Data types:
a. Discrete.
b. Continuous.
2. Counting Techniques:
a. Permutations (‫)التباديل‬
b. Combinations (‫)التوافيق‬.
Introduction to probability (1)
• Definitions:
Permutation is an arrangement of
objects in different orders. There
are basically two types of
permutation.
1. Repetition is allowed.
2. No repetition.
Introduction to probability (1)
A) Permutation with Repetition: here
order is important ( ‫الترتيب مهم والتكرار‬
‫)مسموح‬.
If you have n things to choose from and
you choose r of them then the
permutations are:
n  n  ..........  n ( r times)  n
r
Introduction to probability (1)
Example:
• If there are 6 numbers {1, 2, 3, 4, 5, 6}
and you choose 4 of them with
repetition then we have
6  6  6  6  6  1296
4
permutations
Introduction to probability (1)
B) Permutation without Repetition: here
order is important ( ‫الترتيب مهم والتكرار‬
‫)مسموح‬
• The formula of permutation without
repetition is:
 n
n!
P : n Pr  P(n, r )  P  
 r  (n  r )!
r
n
Introduction to probability (1)
Example:
Suppose we have a “ABCD” litters how
many words we can compose from
them with 4 litters without repetition.
 4
4!
4  3  2 1
P : 4 P 4  P(4,4)  P  

 24 wiords
1
 4  (4  4)!
r
n
Introduction to probability (1)
A
B
C
D
ABCD BACD CABD DABC
ABDC BADC CADB DACB
ACBD BCAD CBAD DBAC
ACDB BCDA CBDA DBCA
ADBC BDAC CDAB DCAB
ADCB BDCA CDBA DCBA
Introduction to probability (1)
• But how many words with 4 litters we
can compose from 4 litters with
repetition?
• Solution:
(4)  256 words
4
Note:
0! = 1
1! = 1
Introduction to probability (1)
• Example:
• If A:={p, q, r, s} find the number of
permutation for 3 elements:
A) Without repetition
 4
4!
4  3  2 1
P : 4 P3  P(4,3)  P  

 24 wiords
1
 3  (4  3)!
r
n
Introduction to probability (1)
• Example:
• If A:={p, q, r, s} find the number of
permutation for 3 elements:
B) With repetition
(4)  64 words
3