Download Probability Rules

Introduction to
Probability & Statistics
Concepts of Probability
Probability Concepts
S = Sample Space : the set of all possible unique
outcomes of a repeatable experiment.
Ex:
flip of a coin
S = {H,T}
No. dots on top face of a die
S = {1, 2, 3, 4, 5, 6}
Body Temperature of a live human
S = [88,108]
Probability Concepts
Event: a subset of outcomes from a sample
space.
Simple Event: one outcome; e.g. get a 3 on one
throw of a die
A = {3}
Composite Event: get 3 or more on throw of a
die
A = {3, 4, 5, 6}
Rules of Events
Union: event consisting of all outcomes present
in one or more of events making up
union.
Ex:
A = {1, 2} B = {2, 4, 6}
A  B = {1, 2, 4, 6}
Rules of Events
Intersection: event consisting of all outcomes
present in each contributing event.
Ex:
A = {1, 2}
A  B = {2}
B = {2, 4, 6}
Rules of Events
Complement: consists of the outcomes in the
sample space which are not in stipulated
event
Ex:
A = {1, 2}
S = {1, 2, 3, 4, 5, 6}
A = {3, 4, 5, 6}
Rules of Events
Mutually Exclusive: two events are mutually
exclusive if their intersection is null
Ex:
A = {1, 2, 3}
AB={ }=
B = {4, 5, 6}
Probability Defined
 Equally
Likely Events
If m out of the n equally likely outcomes in an
experiment pertain to event A, then
p(A) = m/n
Probability Defined
 Equally
Likely Events
If m out of the n equally likely outcomes in an
experiment pertain to event A, then
p(A) = m/n
Ex: Die example has 6 equally likely
outcomes:
p(2) = 1/6
p(even) = 3/6
Probability Defined
 Suppose
we have a workforce which is
comprised of 6 technical people and 4 in
administrative support.
Probability Defined
 Suppose
we have a workforce which is
comprised of 6 technical people and 4 in
administrative support.
P(technical) = 6/10
P(admin) = 4/10
Rules of Probability
Let A = an event defined on the event space S
1.
2.
3.
4.
0 < P(A) < 1
P(S) = 1
P( ) = 0
P(A) + P( A ) = 1
Addition Rule
P(A B) = P(A) + P(B) - P(A  B)
A
B
Addition Rule
P(A B) = P(A) + P(B) - P(A  B)
A
B
Example
 Suppose
we have technical and
administrative support people some of whom
are male and some of whom are female.
Example (cont)
 If
we select a worker at random, compute the
following probabilities:
P(technical) = 18/30
Example (cont)
 If
we select a worker at random, compute the
following probabilities:
P(female) = 14/30
Example (cont)
 If
we select a worker at random, compute the
following probabilities:
P(technical or female) = 22/30
Example (cont)
 If
we select a worker at random, compute the
following probabilities:
P(technical and female) = 10/30
Example (cont)
 Alternatively
we can find the probability of
randomly selecting a technical person or a
female by use of the addition rule.
P (T  F ) = P (T ) + P ( F ) - P (T  F )
= 18/30 + 14/30 - 10/30
= 22/30
Operational Rules
Mutually Exclusive Events:
P(A B) = P(A) + P(B)
A
B
Conditional Probability
Now suppose we know that event A has
occurred. What is the probability of B given A?
A
AB
P(B|A) = P(A  B)/P(A)
Example
 Returning
to our workers, suppose we know
we have a technical person.
Example
 Returning
to our workers, suppose we know
we have a technical person. Then,
P(Female | Technical) = 10/18
Example
 Alternatively,
P(F | T) = P(F  T) / P(T)
= (10/30) / (18/30) = 10/18
Independent Events
 Two
events are independent if
P(A|B) = P(A)
or
P(B|A) = P(B)
In words, the probability of A is in no way
affected by the outcome of B or vice versa.
Example
 Suppose
we flip a fair coin. The possible
outcomes are
H
T
The probability of getting a head is then
P(H) = 1/2
Example
 If
the first coin is a head, what is the
probability of getting a head on the second
toss?
H,H H,T
T,H T,T
P(H2|H1) = 1/2
Example
 Suppose
we flip a fair coin twice. The
possible outcomes are:
H,H H,T
T,H T,T
P(2 heads) = P(H,H) = 1/4
Example
 Alternatively
P(2 heads) = P(H1  H2)
= P(H1)P(H2|H1)
= P(H1)P(H2)
= 1/2 x 1/2
= 1/4
Example
 Suppose
we have a workforce consisting of
male technical people, female technical
people, male administrative support, and
female administrative support. Suppose the
make up is as follows
Tech
Admin
Male
8
8
Female
10
4
Example
Let M = male, F = female, T = technical, and
A = administrative. Compute the following:
Tech
Male
Female
Admin
8
8
10
4
P(M  T) = ?
P(T|F) = ?
P(M|T) = ?