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INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY
Chapter 3: Basic Probability Concepts
3.1 General Definitions and Concepts:
Probability:
Probability is a measure (or number) used to measure the chance of the occurrence of
some event. This number is between 0 and 1.
An Experiment:
An experiment is some procedure (or process) that we do.
Sample Space:
The sample space of an experiment is the set of all possible outcomes of an
experiment. Also, it is called the universal set, and is denoted by Ω.
An Event:
Any subset of the sample space Ω is called an event.
•φ⊆Ω is an event (impossible event)
• Ω⊆Ω is an event (sure event)
Example:
Experiment: Selecting a ball from a box containing 6 balls numbered 1,2,3,4,5 and 6.
•This experiment has 6 possible outcomes
Ω={1, 2, 3, 4, 5, 6}.
•Consider the following events:
E1 =getting an event number = 2, 4, 6 }⊆Ω
E2 =getting a number less than 4 = 1, 2, 3 }⊆Ω
E3 =getting 1 or 3 = 1, 3}⊆Ω
E4 =getting an odd number = 1, 3, 5}⊆Ω
E5 =getting a negative number = }=φ⊆Ω
E6 =getting a number less than 10 = 1, 2,3, 4, 5, 6 }=Ω⊆Ω
Notation: n(Ω =)no. of outcomes (elements) in Ω
n(E =)no. of outcomes (elements) in the event E
INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY
Equally Likely Outcomes:
The outcomes of an experiment are equally likely if the occurrences of the outcomes
have the same chance.
Probability of An Event:
• If the experiment has N equally likely outcomes, then the probability of the
event E is denoted by P(E) and is
defined by:
P(E =)
n E)
=
n(Ω )
n (E )
no. of outcomes in E
=
N
no. of outcomes in Ω
Example:
In the ball experiment in the previous example, suppose the ball is selected randomly.
Determine the probabilities of the following events:
E1 =getting an event number
E2 =getting a number less than 4
E3 =getting 1 or 3
Solution:
}; n Ω)=6
E1 ={2, 4, 6 }; n(E1 )=3
E2 ={1, 2, 3 }; n(E2 )=3
E3 ={1, 3 }; n(E3 )=2
Ω={1, 2, 3, 4, 5, 6
The outcomes are equally likely .
∴P(E1 =)3
P(E1 =)3
6
,
P(E3 =)2 ,
6
3.2 Some Operations on Events:
Let A and B be two events defined on the sample space Ω.
Union of Two events:
A ∪B
A ∪B Consists of all outcomes in A or in B or in both A and B.
A ∪B Occurs if A occurs, or B occurs, or both A and B occur.
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INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY
Intersection of Two Events: A ∩B
A ∩B Consists of all outcomes in both A and B.
A ∩B Occurs if both A and B occur.
Complement of an Event: Ac
Ac is the complement of A.
Ac consists of all outcomes of Ω but are not in A.
Ac occurs if A does not.
Example:
Experiment: Selecting a ball from a box containing 6 balls numbered 1, 2, 3, 4, 5, and
6 randomly.
Define the following events:
E1 ={2, 4, 6 }= getting an even number. E2 ={1, 2,
3}= getting a number< 4.
INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY
E3 ={1, 3 }= getting 1 or 3.
E4 ={1, 3, 5}= getting an odd number.
(1)
E1 ∪E2 ={1, 2, 3, 4, 6 }
= getting an even number or a number less than 4
𝑝(𝐸1 ∪ 𝐸2 ) =
𝑛(𝐸1 ∪ 𝐸2 ) 5
=
6
𝑛( Ω)
(2) E1 ∪E2 ={1, 2, 3, 4, 6 }
= getting an even number or a number less than 4
𝑝(𝐸1 ∪ 𝐸2 ) =
𝑛(𝐸1 ∪ 𝐸4 ) 6
= =1
6
𝑛( Ω)
Note: E1 ∪E4 = Ω. E1 and E4 are called exhaustive events.
INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY
(3)
E1 ∩E2 ={2 }= getting an even number and a number
less than 4.
𝑛(𝐸1 ∪ 𝐸2 ) 1
𝑝(𝐸1 ∩ 𝐸2 ) =
=
𝑛(Ω)
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E1 ∩E2 =Ø= getting an even number and an odd number.
(4)
𝑝(𝐸1 ∩ 𝐸2 ) =
𝑛(𝐸1 ∪ 𝐸4 ) 𝑛(Ø) 0
=
= =0
𝑛(Ω)
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Note: E1 ∩E4 =φ. E1 and E4 are called disjoint (or mutually exclusive) events.
c
E1c = not getting an even number = {2, 4, 6} = {1, 3, 5}
(5)
= getting an odd number.
= E4
Notes:
1.
The events A1 , A2 ,K, An
are exhaustive events if
A1 ∪A2 ∪K∪An =Ω.
2. The events A and B are disjoint (or mutually exclusive) if
A ∩B =φ.
It is impossible that both events occur together (in the same time). In this case:
INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY
 P(A ∩B =)0
 P(A ∪B =)P()A+P B)
If A∩B ≠φ, then A and B are not mutually exclusive (not disjoint).
A∩B ≠φ
A and B are not
mutually exclusive
(It is possible that both
events occur in the same
A∩B = φ
A and B are mutually
exclusive (disjoint)
(It is impossible that both
events occur in the same
time)
time)
3. A ∪Ac =Ω, A and Ac are exhaustive events.
A ∩Ac =φ, A and Ac are disjoint events.
4.
n Ac =n(Ω −)()nA
P Ac =1 −P A)
3.3 General Probability Rules:
1.
0 ≤P(A ≤)1
2.
3.
4.
5.
P(Ω =)1
P(φ =)0
P Ac =1−P(A )
For any events A and B:
6.
P(A ∪B )=P(A +)(PB −)P A ∩B)
For disjoint events A and B
INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY
7.
P(A ∪B)=P(A +)(PB )
For any events A and B:
P(A∩BC)= P(A) −P(A∩B)
P(AC∩B)= P(B) −P(A∩B)
P(AC∩BC)= 1 −P(A∪B)
8.
For mutually exclusive (disjoint) events E1 , E2 ,K, En :
P(E1 ∪E2 ∪K∪En )= P E1 )+P E2 )+L+P En )
3.4 Applications:
We will consider several examples.
Example:
630 patients are classified as follows : (Simple frequency table)
Blood
O
A
B
AB
(E1 ) E2 )
E3 )
E4 )
Type
Total
No. of
284
258
63
25
630
patients
•Experiment: Selecting a patient at random and observe his/her blood type.
•This experiment has 630 equally likely outcomes
∴n(Ω =)630