1. University of Nizwa College of Arts and Sciences
Principles of Probability
Stat-210
Eltayeb Abuelyaman
Spring 2012

2. About Probability Basic concepts Classical Relative Frequency
Properties
Examples

3. Basic Probability Concepts
Probability: Frequently encountered in everyday communication
A physician may say that a patient has a chance of surviving a certain operation.
Another physician may say that she is 95 percent certain that a patient has a particular disease.
Most people express probabilities in terms of percentages.

4. Probability : Fraction
But, it is more convenient to express probabilities as fractions. Thus, we may measure the probability P of the occurrence of some event by a number between 0 and 1: ≤ P ≤1
The more likely the event, the closer the number is to one.
An event that can’t occur has a probability of zero, and an event that is certain to occur has a probability of one.

5. Objective Probability
Classical Probability:
For example, in the rolling of the die, each of the six sides is equally likely to be observed. So, the probability that a 4 will be observed is equal to 1/6.
Equally Likely Outcomes
are the outcomes that have the same chance of occurring.
The set of all possible outcomes (The universal set) S contains N mutually exclusive and equally likely outcomes.
The empty set φ contains no element.

6. The probability of the occurrence of E
The event, E is a set of outcomes in S which has a certain characteristic.
The probability of the occurrence of E
P(E) = m/N, where m is the number of outcomes which satisfy the event E.

7. Subjective Probability
Relative Frequency Probability:
If some process is repeated a large number of times, N, and if some resulting event E occurs m times, the relative frequency of occurrence of E, m/n will be approximately equal to the probability of E P(E) = m/N.
Subjective Probability
Probability measures the confidence that a particular individual has in the truth of a particular proposition.
For example, the probability that a cure for cancer will be discovered within the next 10 years.

8. Given some process (or experiment) with n mutually exclusive events
E1, E2, …, En, then
P (Ei) ≥ 0, i = 1, 2, … n
P (E1) + P (E2) + … + P (En) = 1

9. Properties of Probability
Relations Between Events:
1) Union:
A B means A or B.
Example:
Let S = {1,2,3,4,5,6,7,8,9,10},
Let A be choosing an odd
number > 2,
then A = {3,5,7,9} and P(A) = 0.4
Let B be choosing a number
divisible by 3,
then B = {3,6,9}, P(B) = 0.3.
A  B = {3,5,6,7,9} and P(A  B) = 0.5 10 2
4 8 3 9
5 7 6 B A

10. 2) Intersection: 3) Complement: Example: Example: A  B means A and B.
In the above example, A = {3,5,7,9},
and B = {3,6,9}, then
A  B = {3,9} and P(A  B) = 0.2 B A A 2
4 8 3 9
5 7 6 10
3) Complement:
A` means the complement of A, where
A  A` = S and A  A` = φ.
Example:
In the above example,
B = {3,6,9}, P(B) = 0.3, then
B` = {1,2,4,5,7,8,10} and P(B`) = 0.7. B B A 2
4 8 3 9
5 7 6 10

11. Rules of Probability For example,
1- A and B are called disjoint if A  B = , and then P(A  B) = 0 and P(A  B) = P(A) + P(B).
For example,
if A is choosing an odd number < 11,
A = {1,3,5,7,9} and
B is choosing an even number < 11,
B = {2,4,6,8,10}.
Then P(A  B) = 0 and P(A  B) = P(A) + P(B) = 1.
2- If A and B are not disjoint, then
P(A  B) = P(A) + P(B) - P(A  B)
if A is choosing a number divisible by 5
A = {5,10} and
Then P(A  B) = 0.1 and
P(A  B) = P(A) + P(B) - P(A  B) = 0.6. A B B A 1 3
7 9 5

12. For example, 3- P(A) + P(A`) = 1. 4- P(A) = P(A  B) + P(A  B`)
if A is choosing a number < 5,
A = {1,2,3,4}, P(A) = 0.4, but
A` = {5,6,7,8,9,10}, P(A’) = 0.6
Then P(A) + P(A`) = 1.
4- P(A) = P(A  B) + P(A  B`)
if A is choosing a number divisible by 5
A = {5,10} , P(A) = 0.2 and
B is choosing an even number < 11,
B = {2,4,6,8,10}.
Then P(A  B) = 0.1 and
P(A  B`) = 0.1
Then P(A) = P(A  B) + P(A  B`) A
5 6 7
8 9 10 B A 1 3
7 9 5

13. Two- way Table of Probabilities:
5- P(A`  B`) = 1 – P(A  B).
For example,
if A is choosing a number divisible by 5
A = {5,10} , P(A) = 0.2 and
B is choosing an even number < 11,
B = {2,4,6,8,10}.
Then P(A  B) = 0.6 and
P(A`  B`) = 0.4.
Then P(A`  B`) = 1 – P(A  B).
Two- way Table of Probabilities: 1 3
7 9 A B 5
Total B` B
P(A)
P(A  B`)
P(A  B) A
P(A`)
P(A`  B`)
P(A`  B) A`
P(S) = 1
P(B`)
P(B)

14. Calculating The Probability of an Event Example: Here is the data of a sample of adults in a certain city:
Sum
Female
(B`)
Male
(B) 23 8 15
Diabetic
(A)
102 62 40
Normal
(A`)
125 70 55
P(A) = 23 / 125
P(A`) = 1- P(A) = 102 / 125
P(B) = 55 / 125
P(B`) = 70 / 125
P(A  B) = 15 / 125
P(A  B`) = 8 / 125
P(A` B) = 40 / 125
P(A`  B`) = 62 / 125
P(A) = P(A  B) + P(A  B`) =
23 / 125
P(A  B) = P(A) + P(B) - P(A  B) = 23 / / 125 – 15 / 125 =
= 63/125 B 15 A 40 8 15 62

15. Example: For a sample of 80 recently born children, the following table is obtained:
P(N) = 57/80
P(O) = 16/80
P(B) = 49/80
P(G) = P(B`) = 1 - P(B) = 31/80
P(L  B) = 4/80
P(N  B) = 35/80
P(O  B) = 10/80
P(L  G) = 3/80
P(N  G) = 22/80
P(O  G) = 6/80
P(L  B) = 7/ /80 - 4/80 =
= 52/80
P(N  B) = 57/ / /80 =
= 71/80
P(O  B) = 16/ / /80 =
= 55/80
P(L  N) = 7/ /80 = 64/80
P(B  G) = 49/ /80 = 1
Sum
Girl
(G)
Boy
(B)
Sex
Weight in Kg. 7 3 4
< 2.5 Kg
(L) 57 22 35
2.5 - < 3
(N) 16 6 10
3 +
(O) 80 31 49