basic statistics

1. Chapter Four
Probability and Probability distribution
Introduction to probability
The word probability has two basic meaning
(i) a quantitative measure of uncertainty
(ii) measure degree of belief in a particular statement or problem
Example: predictions are based on probability, and hypotheses are tested by using probability
• which is concerned with decision making under risk and uncertainty conditions or computing the
chance that something will happen in the future.
• Probability is the chance of an event occurring
• it is the measure of how likely an outcome is occurs

2. Definitions of Some Probability Terms
• Experiment: Any process of observation or measurement or any process which generates well
defined outcome.
• Probability Experiment (Random Experiment): it is an experiment which can be repeated any
number of times under the same conditions, but their outcomes are uncertain and does not give
unique results.
• Outcome: The result of a single trial of a random experiment
Example: Experiment Outcome
Rolling a fair die Head, Tail
Tossing a fair coin 1, 2, 3, 4, 5, 6.

3. Cont..
• Sample space: a collection/set of all possible outcomes of an experiment
• Event: Any subset of sample space.
• Equally Likely Events: Events which have the same chance of occurring
• Elementary event: An event consisting a single out come.
• Impossible event: An event which can’t occur.
 Example: Considering the experiment of rolling of a die let A be the event of odd numbers,
B be the event of even numbers, and C be the event of number 8.
𝑆 = 1, 2, 3, 4, 5, 6
𝐴 = 1, 3, 5
𝐵 = 2, 4, 6
𝐶 = impossible event

4. Cont..
• Complement of an event: The complement of event 𝐴 (denoted by 𝐴′ or 𝐴𝑐 ), consists of all the sample
points in the sample space that not in A.
I. 𝑃 𝐴 + 𝐴′ = 𝑆
II. 𝑃 𝐴 + 𝑃 𝐴′
= 1
III. 𝑃 𝐴 = 1 − 𝑃(𝐴′)
• Mutually Exclusive(Disjoint) Events: Two events which can not happen at the same time
Example: Considering the experiment of rolling of a die let 𝐸1 be the event of odd numbers, 𝐸2 be the
event of even numbers.
𝑆 = 1, 2, 3, 4, 5, 6
𝐸1 = 1, 3, 5 𝐸2 = 2, 4, 6 P 𝐸1 ∩ 𝐸2 = 𝜙 , therefore the two event are mutual exclusive.

5. Cont..
 Independent events: Two events are said to be independent ,if the occurrence of one is
not affected by, and does not affect, the other. If two events are not independent, then they
are said to be dependent.
 For example: if two fair coins are tossed, then the result of one toss is totally independent
of the result of the other toss.
 Any event consisting of a single point of the sample space is known as a simple event
 if any event consists of more than one single point of the sample space then such an event
is called a compound event.

6. Set Theory
 Set is a collection of well-defined objects. These objects are called elements.
• Sets usually denoted by capital letters and elements by small letters.
• Membership for a given set can be denoted by ∈ to show belongingness
 Set can be finite or infinite
Countable finite set: is a set which contains a finite number of elements.
E.g. let set A = {1, 2, …….,100}, and {x: x is an integer 0< x <5}
Countable infinite set: is a set which contains an infinite number of elements.
E.g. let set B = {1, 2, 4, 5,……….}
Uncountable infinite: a set that is not countable infinitely.
E.g.{x: x ∈ ℜ, 0 < x < 5}, {𝑥: 𝑥 ≥ 1} and also {x : x ∈ ℜ , x > 0}

7. Cont..
Types of set
Universal set: is a set that contains all elements of the set that can be
considered the objects of that particular discussion.
Empty or null set: is a set which has no element, denoted by {} or 𝜙
Sub set: If every element of set A is also elements of set B, set A is called
sub sets of B, and denoted by A ⊆ 𝐵
Proper subset: For two sets A and B if A is subset of B and B is not sub set
of A, then A is said to be a proper subset of B. Denoted by A ⊂ 𝐵

8. Cont..
 If A is a proper subset of B, then all element of A are in B but set B contains at least one
element not in A.
 NB: every proper subset can be subset but the invers may not be always true.
Example: let set A = {1, 2, 3} and B = {1, 2, 3, 5, 7, 9, 11}
Then A ⊆ 𝐵 and A ⊂ 𝐵
 let set A = {1, 2, 3} and B = {1, 2, 3}then A⊆ 𝐵 but not proper subset.

9. Set operation
• Union of sets: The union of two sets A and B is a set which contains elements
which belongs to either of the two sets. Union of two sets denoted by ∪, A ∪ 𝐵 (A
union B). the set with all elements in A or B or both.
• Intersection of sets: The intersection of two sets A and B is a set which contains
elements which belongs to both sets A and B. Intersection of two sets denoted by ∩
, 𝐴 ∩ 𝐵 (A intersection B).
• Disjoint sets: are two sets whose intersection is empty set.
𝐴 ∩ 𝐵 = 𝜙

10. Properties of set operation
1. 𝐴 ∪ ∅ = 𝐴 𝑎𝑛𝑑 𝐴 ∩ ∅ = ∅
2.
𝐴 ∪ 𝐵 ′ = 𝐴′ ∩ 𝐵′
𝐴 ∩ 𝐵 ′
= 𝐴′
∪ 𝐵′ De Morgan's Rule

11. Cont..
Some similarities between notions in set theory and that of probability
theory are
In probability theory In set theory
Event A or event B 𝐴 ∪ 𝐵
Event A and event B 𝐴 ∩ 𝐵
Event A is impossible 𝐴 = ∅
Event A and B are mutual exclusive 𝐴 ∩ 𝐵 = ∅

12. Counting rule
• In order to calculate probabilities, we have to know
1. The number of elements of an event
2. The number of elements of the sample space
• In order to determine the number of outcomes, one can use several rules of counting.
1. The addition rule
2. The multiplication rule
3. Permutation rule
4. Combination rule

13. Addition rule
• Addition Rule: If event 𝐴 can occur in 𝑚 possible ways and event 𝐵 can occur in 𝑛 possible ways,
there are 𝑚 + 𝑛 possible ways for either event 𝐴 or event 𝐵 to occur, but no events in common between
them.
n A + B = n(A) + n(B) − n(A ∩ B)
• Example: A student goes to the nearest cafe to have a breakfast. He can take tea, coffee, or milk with
bread, cake and sandwich. How many possibilities does he have?
• Therefore there are nine possibility
Tea
Bread
Cake
sandwich
Coffee
Bread
Cake
sandwich
Milk
Bread
Cake
sandwich

14. The multiplication Rule
• If a choice consists of 𝑘 steps of which the first can be made in 𝑛1 ways, the second can be made in 𝑛2 ways, the 𝑘𝑡ℎ
can be made in 𝑛𝑘 ways then the whole choice can be made in 𝑛1 × 𝑛2 × 𝑛3 … … . 𝑛𝑘 ways.
• multiply the number of possible outcomes for each event by the number of possible outcomes of the other events
• Example 1 :A student has two shoes, three trousers and three jackets. In how many can be dressed
Solution: he can select shoes by 2 ways, trouser by 3 ways, and jacket by 3 ways.
number of ways to select different outfits = 2*3*3 = 18 ways
• Example 2 : The digits 0, 1, 2, 3, and 4 are to be used in 4 digit identification card. How many different cards are
possible if
i. Repetitions are permitted.
ii. Repetitions are not permitted.

15. Conti…..
Solution:
There are four steps:
• Selecting the 1st digit, this can be made in 5 ways
• Selecting the 2nd digit, this can be made in 5 ways
• Selecting the 3rd digit, this can be made in 5 ways
• Selecting the 4th digit, this can be made in 5 ways
5*5*5*5 = 625 different card are possible.
1st
digit 2nd digit 3rd digit 4th digit
5 5 5 5

16. Conti…
Solution:
There are four steps:
• Selecting the 1st digit, this can be made in 5 ways *the 1st object can be chosen 𝑛 way
• Selecting the 2nd digit, this can be made in 4 ways *the 1st object can be chosen 𝑛 − 1 way
• Selecting the 3rd digit, this can be made in 3 ways
• Selecting the 4th digit, this can be made in 2 ways
5*4*3*2 = 120 different card are possible
1st digit 2nd digit 3rd digit 4thdigit
5 4 3 2

17. Permutation
• Permutation is an arrangement of 𝑛 different objects in a specified order
Permutation rule:
1. The number of permutations of 𝑛 distinct objects taken all together is 𝑛!
Where 𝑛! = 𝑛 ∗ 𝑛 − 1 ∗ 𝑛 − 2 ∗ ⋯
Note: by definition 0! = 1
Anything permute itself is equivalent to itself factorial
2. The arrangement of 𝑛 objects in a specified order using 𝑟 objects at a time is called permutation of 𝑛 objects taken 𝑟 objects
at a time
It is written as 𝑛𝑃𝑟 and the formula is
𝑛𝑃𝑟 =
𝑛!
𝑛−𝑟 !
3. permutation where some objects are identical or an ordered partition of 𝑛 objects into 𝑘 groups of sizes 𝑛1, 𝑛2, · · · , 𝑛𝑘 is
given
𝑛
𝑛1 , 𝑛2, … . 𝑛𝑘
=
𝑛!
𝑛1!∗𝑛2!….𝑛𝑘!

18. Examples:
1. Suppose we have a letters A,B, C, D
a) How many permutations are there taking all the four?
b) How many permutations are there if two letters are used at a time?
2. How many different permutations can be made from the letters in the word
“CORRECTION”?

19. Combination
• combination is a selection of all or part of a set of objects, without regard to the order in which they were
selected.
Combination rule
• The number of combinations of 𝑟 objects selected from 𝑛 object is denoted by:
• 𝑛𝐶𝑟 =
𝑛!
𝑛−𝑟 ! 𝑟!

22. Approaches to Measuring Probability
• There are three different conceptual approaches to study probability theory.
1. The classical approach
2. The frequentist approach
3. The axiomatic approach
Classical approach
• If the random experiment with 𝑁 equally likely outcomes is conducted and out of these 𝑁𝐴
outcome are favorable to the event 𝐴, then the probability that event 𝐴 occur denoted 𝑃(𝐴) is
defined as:
P A =
NA
N
=
No.of outcomes favorable to A
Total number of outcomes
=
n(A)
n(S)
• All outcomes are equally likely and mutually exclusive
• Total number of outcome is finite, say 𝑁

25. Cont..
Limitation
• If it is not possible to enumerate all the possible outcomes for an experiment
• If the sample points (outcomes) are not mutually independent
• If the total number of outcomes is infinite
• If each and every outcomes is not equally likely

26. The Frequentist Approach
• Relative frequency probability: If some process is repeated a large number of 𝑛 times, and some resulting
event 𝐸 occurs 𝑚 times, Therefore the probability of the event 𝐸 happening in the long run is given by:
• 𝑃 𝐸 =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑒𝑣𝑒𝑛𝑡 𝐸 ℎ𝑎𝑠 𝑜𝑐𝑐𝑢𝑟𝑒𝑑
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
=
𝑚
𝑛

28. Axiomatic approach
 For a given experiment, 𝑺 denotes as ample space associated with a random experiment, then a function
which assign every event 𝑨 ∈ 𝑺 to unique non-negative real number 𝑷(𝑨) is called probability function, if
the following holds
i. 𝑃 𝐴 ≥ 0
ii. 0 ≤ 𝑃(𝐴) ≤ 1
iii. 𝑃 𝑆 = 1
iv. If 𝐴 and 𝐵 are mutual exclusive events, P 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃(𝐵)
v. 𝑃 𝐴′
= 1 − 𝑃(𝐴)
vi. 𝑃 ∅ = 0; ∅ is impossible event
vii. Two event are not mutual exclusive: P 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)

29. Conditional probability and independent event
 Conditional events: If the occurrence of one event has an effect on the next occurrence of
the other event then the two events are conditional or dependent events.
• The conditional probability of an event 𝐴 given that 𝐵 has already occurred, denoted by
𝑃 𝑃𝐴/𝐵 =
𝑃(𝐴 ∩𝐵)
𝑃(𝐵)
, 𝑃 𝐵 ≠ 0
• Remark:
1. P 𝐴′/𝐵 = 1 − 𝑃 𝐴/𝐵
2. P 𝐵′
/𝐴 = 1 − 𝑃(𝐵/𝐴)

32. Probability of independent events
• Two event A and B are independent, if and only if
1. P 𝐴 ∩ 𝐵 = 𝑃 𝐴 ∗ 𝑃 𝐵
2. 𝑃 𝐴/𝐵 = 𝑃(𝐴)
3. 𝑃 𝐵/𝐴 = 𝑃(𝐵)

33. Total Probability rules & Bayes’ theorem
Total probability rule more than two event
• From the figure. The event A1, A2, A3, A4, A5 are mutually exclusive and S = A1+A2+A3+A4+A5 this
event are said to form a partition of the sample.
• If B is the event defined on the sample space S, then for any event B defined on S such that P(B)>0,
Then 𝑩 = 𝑨𝟏 ∩ 𝑩 ∪ 𝑨𝟐 ∩ 𝑩 , … , (𝑨𝟓 ∩ 𝑩)

34. Cont..
• Since (𝐴1 ∩ 𝐵) , (𝐴2 ∩ 𝐵), … . (𝐴5 ∩ 𝐵) are mutually exclusive events
𝑃(𝐵) = 𝑃(A1 ∩ B) + P(A2 ∩ B)+ ….. + (A5 ∩ B).
𝑃(𝐴1)𝑃(𝐵/𝐴1 + 𝑃(𝐴2)𝑃(𝐵/𝐴2 + ⋯ 𝑃(𝐴5)𝑃(𝐵/𝐴5.
• Generally: if 𝐴1, 𝐴2, … . . 𝐴𝑛 from partition of a sample space 𝑆, than
for any event 𝐵 defined on 𝑆 such that 𝑃(𝐵) ≥ 0
𝑷 𝑩 = (𝑨𝒊)𝑷(𝑩/𝑨𝒊)

35. Cont..
Bayes theorem
• Let 𝐴1, 𝐴2, … . . 𝐴𝑛 be a collection of events which partition a sample
space S. let B be an event defined on S such that P(B)≠ 0,Then for any
of the events 𝐴𝑗(j=1,2,3,….,n)
• 𝑃 𝑨𝒋/𝑩 =
𝑷 𝑨𝒋 𝑷(𝑩/𝑨𝒋)
𝒊=𝟏
𝒏 𝑷 𝑨𝒋 𝑷 𝑷(𝑩/𝑨𝒋)