Download Report

Probability
Part 1
A Few Terms
Probability represents a (standardized)
measure of chance, and quantifies
uncertainty.
 Let S = sample space which is the set of all
possible outcomes.
 An event is a set of possible outcomes that
is of interest.
 If A is an event, then P(A) is the probability
that event A occurs.

L. Wang, Department of Statistics
University of South Carolina; Slide 2
Identify the Sample Space

What is the chance that it will rain today?

The number of maintenance calls for an
old photocopier is twice that for the new
photocopier. What is the chance that the
next call will be regarding an old
photocopier?

If I pull a card out of a pack of 52 cards, what is
the chance it’s a spade?
L. Wang, Department of Statistics
University of South Carolina; Slide 3
Union and Intersection of Events

The intersection of events A and B refers
to the probability that both event A and
event B occur.
P( A  B)

The union of events A and B refers to the
probability that event A occurs or event B
occurs or both events, A & B, occur.
P( A  B)
L. Wang, Department of Statistics
University of South Carolina; Slide 4
Mutually Exclusive Events

Mutually exclusive events can not occur
at the same time.
S
Mutually Exclusive Events
S
Not Mutually Exclusive
Events
L. Wang, Department of Statistics
University of South Carolina; Slide 5

A manufacturer of front lights for
automobiles tests lamps under a high
humidity, high temperature environment
using intensity and useful life as the
responses of interest. The following table
shows the performance of 200 lamps.
L. Wang, Department of Statistics
University of South Carolina; Slide 6
Probability of the Union of Two Events
What is the probability
that a randomly chosen
Good Sat
Unsat Total
light will have performed
Good in Useful Life?
100 25
5 130  Good in Intensity?
Useful
Life
Inten
Good

Sat
35
10
5
50
Unsat
10
8
2
20
145
43
Total

Good in Useful Life or
Good in Intensity?
12 200
L. Wang, Department of Statistics
University of South Carolina; Slide 7
The Union of Two Events

If events A & B intersect, you have to
subtract out the “double count”.
P( A  B)  P( A)  P( B)  P( A  B)

If events A & B do not intersect (are
mutually exclusive), there is no “double
count”.
P( A  B)  P( A)  P( B)
L. Wang, Department of Statistics
University of South Carolina; Slide 8
Useful
Life

Inten
Good
Good
100
25
5 130
35
10
5
Sat
Unsat
Total
Sat
Unsat
10
8
145
43
2
Total
What is the probability
that a randomly chosen
light will have performed
Good in Intensity or
Satisfactorily in Useful
life?
50  130/20
20
A.
B.
12 200 C.
43/200
173/200
148/200
L. Wang, Department of Statistics
University of South Carolina; Slide 9
Useful
Life

Inten
Good
Good
100
25
5 130
35
10
5
Sat
Unsat
Total
Sat
Unsat
10
8
145
43
2
Total
What is the probability
that a randomly chosen
light will have performed
Unsatisfactorily in both
useful life and intensity?
50 A. 2/20
20
B.
C.
12 200 D.
32/200
2/200
4/200
L. Wang, Department of Statistics
University of South Carolina; Slide 10
Conditional Probability
Useful
Life
Inten
Good
Good
100
25
5
Sat
35
10
5
Unsat
10
8
2
145
43
12
Total
What is the probability
that a randomly chosen
Total
light performed Good in
Useful Life?
130  Good in Intensity.
 Given that a light had
50
performed Good in Useful
Life, what is the
20
probability that it
performed Good in
200
Intensity? L. Wang, Department of Statistics

Sat
Unsat
University of South Carolina; Slide 11
Conditional Probability
Useful
Life
Inten
Good
Good
100
25
5
Sat
35
10
5
Unsat
10
8
2
145
43
12
Total
Given that a light had
performed Good in
Total
Intensity, what is the
probability that it will
130
perform Good in Useful
Life?
50
A. 100/145
20 B. 100/130
C. 100/200
200

Sat
Unsat
L. Wang, Department of Statistics
University of South Carolina; Slide 12
Useful
Life
A.
5/12
5/130
5/200
10/145
Good
Good
100
25
5 130
35
10
5
50
10
8
2
20 B.
Unsat
Total
145
Unsat
Given that a light had
performed Good in
Intensity, what is the
probability that it
performed
Unsatisfactorily in Useful
life?
Inten
Sat
Sat

43
Total
12 200
C.
D.
L. Wang, Department of Statistics
University of South Carolina; Slide 13
Conditional Probability

The conditional probability of B, given
that A has occurred:
P( A  B)
P( B | A) 
P( A)
L. Wang, Department of Statistics
University of South Carolina; Slide 14
Probability of Intersection

Solving the conditional probability formula
for the probability of the intersection of A
and B:
P( A  B)
P( B | A) 
P( A)
P( A  B)  P( A)  P( B | A)
L. Wang, Department of Statistics
University of South Carolina; Slide 15
We purchase 30% of our parts from
Vendor A. Vendor A’s defective rate is
5%. What is the probability that a
randomly chosen part is defective and
from Vendor A?
0.200
B. 0.050
C. 0.015
D. 0.030
A.
L. Wang, Department of Statistics
University of South Carolina; Slide 16
We are manufacturing widgets.
50% are red, 30% are white
and 20% are blue. What is the
probability that a randomly
chosen widget will not be
white?
A. 0.70
B. 0.50
C. 0.20
D. 0.65
L. Wang, Department of Statistics
University of South Carolina; Slide 17
When a computer goes down, there is a
75% chance that it is due to an overload
and a 15% chance that it is due to a
software problem. There is an 85%
chance that it is due to an overload or a
software problem. What is the probability
that both of these problems are at fault?
A. 0.11
B. 0.90
C. 0.05
D. 0.20
L. Wang, Department of Statistics
University of South Carolina; Slide 18
It has been found that 80% of all
accidents at foundries involve human
error and 40% involve equipment
malfunction. 35% involve both
problems. If an accident involves an
equipment malfunction, what is the
probability that there was also human
error?
A. 0.3200
B. 0.4375
C. 0.8500
D. 0.8750
L. Wang, Department of Statistics
University of South Carolina; Slide 19
Suppose there is no Conditional Relationship
between Useful Life & Intensity.
Useful
Life
Inten
Good
Good
128
16
Sat
16
2
2
20
Unsat
16
2
2
20
160
20
Total
Sat
Unsat

What is the probability a
light performed Good in
Intensity?

Given that a light had
performed Good in Useful
Life, what is the
probability that it will
perform Good in
Intensity?
Total
16 160
20 200
L. Wang, Department of Statistics
University of South Carolina; Slide 20
When P( B | A)  P( B) , We Say that
Events B and A are Independent.
The basic idea underlying independence is
that information about event A provides no
new information about event B. So “given
event A has occurred”, doesn’t change our
knowledge about the probability of event B
occurring.
L. Wang, Department of Statistics
University of South Carolina; Slide 21
 There
are 10 light bulbs in a bag, 2
are burned out.
 If we randomly choose one and test
it, what is the probability that it is
burned out?
 If we set that bulb aside and
randomly choose a second bulb, what
is the probability that the second bulb
is burned out?
L. Wang, Department of Statistics
University of South Carolina; Slide 22
Near Independence

EX: Car company ABC manufactured
2,000,000 cars in 2008; 1,500,000 of the
cars had anti-lock brakes.
– If we randomly choose 1 car, what is the
probability that it will have anti-lock
brakes?
– If we randomly choose another car, not
returning the first, what is the
probability that it will have anti-lock
brakes?
L. Wang, Department of Statistics
University of South Carolina; Slide 23
Independence

Sampling with replacement makes
individual selections independent from
one another.

Sampling without replacement from
a very large population makes
individual selection almost independent
from one another
L. Wang, Department of Statistics
University of South Carolina; Slide 24
Probability of Intersection

Probability that both events A and B occur:
P( A  B)  P( A)  P( B | A)

If A and B are independent, then the
probability that both occur:
P( A  B)  P( A)  P( B)
L. Wang, Department of Statistics
University of South Carolina; Slide 25
Test for Independence


If P( B | A)  P( B) , then A and B are
independent events.
If A and B are not independent events,
they are said to be dependent events.
L. Wang, Department of Statistics
University of South Carolina; Slide 26
Four electrical components are connected in
series. The reliability (probability the
component operates) of each component is
0.90. If the components are independent of
one another, what is the probability that the
circuit works when the switch is thrown?
A
A. 0.3600
B
B. 0.6561
C
D
C. 0.7290
D. 0.9000
L. Wang, Department of Statistics
University of South Carolina; Slide 27
Complementary Events
The complement of an event is every
outcome not included in the event, but still
part of the sample space.
 The complement of event A is denoted A.
 Event A is not event A.

S:
A
A
P( A)  P( A )  1
P( A)  1  P( A )
L. Wang, Department of Statistics
University of South Carolina; Slide 28
Mutually exclusive events are
always complementary.
True
B. False
A.
L. Wang, Department of Statistics
University of South Carolina; Slide 29
An automobile manufacturer gives a 5year/75,000-mile warranty on its drive
train. Historically, 7% of the
manufacturer’s automobiles have required
service under this warranty. Consider a
random sample of 15 cars.
 If we assume the cars are independent of
one another, what is the probability that
no cars in the sample require service
under the warrantee?
 What is the probability that at least one
car in the sample requires service?

L. Wang, Department of Statistics
University of South Carolina; Slide 30
Consider the following electrical circuit:
0.95
0.95
0.95
The probability on the components is their
reliability (probability that they will
operate when the switch is thrown).
Components are independent of one
another.
 What is the probability that the circuit will
not operate when the switch is thrown?

L. Wang, Department of Statistics
University of South Carolina; Slide 31
Probability Rules
0 < P(A) < 1
2) Sum of all possible mutually exclusive
outcomes is 1.
3) Probability of A or B:
1)
P( A  B)  P( A)  P( B)  P( A  B)
4)
Probability of A or B when A, B are mutually
exclusive:
P( A  B)  P( A)  P( B)
L. Wang, Department of Statistics
University of South Carolina; Slide 32
Probability Rules Continued
4)
Probability of B given A:
5)
Probability of A and B:
P( A  B)
P( B | A) 
P( A)
P( A  B)  P( A)  P( B | A)
6)
Probability of A and B when A, B are
independent:
P( A  B)  P( A)  P( B)
L. Wang, Department of Statistics
University of South Carolina; Slide 33
Probability Rules Continued
7)
If A and B are compliments:
P( A)  P( A )  1
or
P( A)  1  P( A )
L. Wang, Department of Statistics
University of South Carolina; Slide 34
Consider the electrical circuit below.
Probabilities on the components are
reliabilities and all components are
independent. What is the probability that
the circuit will work when the switch is
thrown?
A
0.90
B
0.90
C
0.95
L. Wang, Department of Statistics
University of South Carolina; Slide 35
The number of maintenance calls for an
old photocopier is twice that for the new
photocopier.
Outcomes
Old Machine
New Machine
Probability
0.67
0.33
Which of the following series of events would most
cause you to question the validity of the above
probability model?
A. Maintenance Call for Old Machine.
B. Maintenance Call for New Machine.
C. Two maintenance calls in a row for old machine.
D. Two maintenance calls in a row for L.new
machine
Wang, Department of Statistics
University of South Carolina; Slide 36