Probability

1
Probability

• Part 1

2
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.

3
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 its a spade?

4
Union and Intersection of Events

• The intersection of events A and B refers to the
  probability that both event A and event B occur.
• 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.

5
Mutually Exclusive Events

• Mutually exclusive events can not occur at the
  same time.

S
S
Mutually Exclusive Events
Not Mutually Exclusive Events
6

• 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.

7
Probability of the Union of Two Events

• What is the probability that a randomly chosen
  light will have performed Good in Useful Life?
• Good in Intensity?
• Good in Useful Life or Good in Intensity?

8
The Union of Two Events

• If events A B intersect, you have to subtract
  out the double count.
• If events A B do not intersect (are mutually
  exclusive), there is no double count.

9

• What is the probability that a randomly chosen
  light will have performed Good in Intensity or
  Satisfactorily in Useful life?
• 130/20
• 43/200
• 173/200
• 148/200

10

• What is the probability that a randomly chosen
  light will have performed Unsatisfactorily in
  both useful life and intensity?
• 2/20
• 32/200
• 2/200
• 4/200

11
Conditional Probability

• What is the probability that a randomly chosen
  light performed Good in Useful Life?
• Good in Intensity.
• Given that a light had performed Good in Useful
  Life, what is the probability that it performed
  Good in Intensity?

12
Conditional Probability

• Given that a light had performed Good in
  Intensity, what is the probability that it will
  perform Good in Useful Life?
• 100/145
• 100/130
• 100/200

13

• Given that a light had performed Good in
  Intensity, what is the probability that it
  performed Unsatisfactorily in Useful life?
• 5/12
• 5/130
• 5/200
• 10/145

14
Conditional Probability

• The conditional probability of B, given that A
  has occurred

15
Probability of Intersection

• Solving the conditional probability formula for
  the probability of the intersection of A and B

16
We purchase 30 of our parts from Vendor A.
Vendor As defective rate is 5. What is the
probability that a randomly chosen part is
defective and from Vendor A?

• 0.200
• 0.050
• 0.015
• 0.030

17
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

18
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

19
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

20
Suppose there is no Conditional Relationship
between Useful Life Intensity.

• 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?

21
When , 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, doesnt change our knowledge about the
probability of event B occurring.
22

• 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?

23
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?

24
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

25
Probability of Intersection

• Probability that both events A and B occur
• If A and B are independent, then the probability
  that both occur

26
Test for Independence

• If , then A and B are
  independent events.
• If A and B are not independent events, they are
  said to be dependent events.

27
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
B
C
D
A. 0.3600 B. 0.6561 C. 0.7290 D.
0.9000
28
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.

• 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
29
Mutually exclusive events are always
complementary.

• True
• False

30

• An automobile manufacturer gives a
  5-year/75,000-mile warranty on its drive train.
  Historically, 7 of the manufacturers
  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?

31
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?

32
Probability Rules

• 0 lt P(A) lt 1
• Sum of all possible mutually exclusive outcomes
  is 1.
• Probability of A or B
• Probability of A or B when A, B are mutually
  exclusive

33
Probability Rules Continued

• Probability of B given A
• Probability of A and B
• Probability of A and B when A, B are independent

34
Probability Rules Continued

• If A and B are compliments

or
35
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
C 0.95
B 0.90
36
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?

• Maintenance Call for Old Machine.
• Maintenance Call for New Machine.
• Two maintenance calls in a row for old machine.
• Two maintenance calls in a row for new machine