Transcript Basic Concepts of Probability

Basic Concepts of Probability
Probability Experiment: an action,or trial through which
specific results are obtained.
Results of a single trial is an outcome
The set of all possible outcomes is the sample space.
All probabilities should be between 0 and 1.
Sets and Venn Diagrams
A Venn Diagram usually consists of a rectangle which
represents the sample space, and circles within it representing
particular events.
6
The event A = {1,2} when rolling a
die.
3
1
A
The sample space S = {1,2,3,4,5,6}
2
4
5
S
Set Notation
S, the sample space is represented by a rectangle
and A, an event, is represented by a circle.
A
S
Ais the complement of A.
Complement of event: The set of all outcomes
in samples space that are not included in event
E. The complement of event E is denoted by
E’ and is read as “E prime”
P( E )  P( E ' )  1
P( E )  1  P( E ' )
P( E ' )  1  P( E )
E
E’
5
2
6
1
3
4
If S = {1,2,3,4,5,6,7}, and A = {2,4,6} then
A
= {1,3,5,7}
x  A reads " x is in A".
i.e, x is an element of set A
n(A) reads ‘the number of elements in set A’
Union
A  B denotes the union of sets A and B This set contains
all elements belonging to A or B, or both A and B.
A  B  {x : x  A or x  B}
A  B is shaded
A
B
Intersection
A B
denotes the intersection of sets A and B. This is the
set of all elements common to both sets.
A  B  {x : x  A and x  B}
A  B is shaded
A
B
Mutually Exclusive
•Disjoint sets are sets which do not have elements in
common
These two sets are disjoint
A
B
A  B  0 represents an empty set.
A and B are said to be mutually exclusive.
Tree Diagram:one way to list
1 H1
2 H2
outcomes.
H
T
Tree Diagram for Coin and
Die Experiment
3
4
5
6
H3
H4
H5
H6
1
2
3
4
5
6
T1
T2
T3
T4
T5
T6
Simple Events
An event that consists of a single outcome
• The outcome {3,2}is
different from the
outcome{2,3}for a
statistician, but not for
a player
• The event {3,2} is a
simple event.
• The event “roll a sum
of 5” is not simple
because it consists of
the four outcomes
{1,4}, {2,3}, {3,2}
and {4,1}.
Law of Large Numbers
As an experiment is repeated over and over, the
empirical probability of an event approaches the
theoretical (actual) probability of the event.
Type
Summary
Formula
Classical
(Theoretical)
Probability
The number of outcomes in
the sample space is known
and each outcome is equally
likely to occur.
P(E)= Number of outcomes in an event E
Number of outcomes in sample space
Empirical
(Statistical)
Probability
The frequency of outcomes in
the sample space is estimated
from experimentation.
P(E)=Frequency of E
Total frequency
=
Subjective
Probability
Probabilities result from
intuition, educated guesses, and
estimates.
None
f
n