G. Baker, Department of Statistics

1
?????

• What is the probability of tossing a head with a
  fair coin?
• What is the probability of tossing 2 heads with
  two tosses of a fair coin?
• What is the probability of tossing at least 2
  heads with three tosses of a fair coin?

2
?????

• What is the probability that at least 2 people in
  this class (n39) have the same birthday Month
  and day?
• Year has 365 days forget leap year.
• Equal likelihood for each day

3
Probability

• A Brief Look

4
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.

5
Distribution of Defects for Extruded Molding
6
Distribution for Life of Light Bulbs
2500,2600), 2600,2700),..3000,3100
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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 outcome, then P(A) is the
  probability that event outcome A occurs.

8
ID the Sample Space, As and P(A)s

• 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?

9
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.

and
either/or
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Mutually Exclusive Events

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

S
S
Mutually Exclusive Events
Not Mutually Exclusive Events
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Roommate profile Distribution
Frequency - Counts
Relative Frequency - Probability
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What is the probability that a randomly chosen
roommate will snore?
13
What is the probability that a randomly chosen
roommate will like to party?
14
What is the probability that a randomly chosen
roommate will snore or like to party?
15
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.

16
Given that a randomly chosen roommate snores,
what is the probability that he/she likes to
party?
17
Conditional Probability

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

Given
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Probability of Intersection

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

19
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.
20

• 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?

21
Near Independence

• EX Car company ABC manufactured 2,000,000 cars
  in 2009 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?

22
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

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Probability of Intersection

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

24
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.

25
Are snoring or not and partying or not
independent of one another?
26
Arrange the counts so that snoring and partying
are independent of one another.
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.

• 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
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All mutually exclusive events are complementary.
?????????????????

• True
• False

29
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

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Probability Rules Continued

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

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Probability Rules Continued

• If A and A are compliments

or
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So Lets Apply the Rules

• 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?

33
????????????

• 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?

34
????????????

• When a computer goes down, there is a 75 chance
  that it is due to 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?

35
????????????

• 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?

36
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
37

• 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 warranty?
• What is the probability that at least one car in
  the sample requires service?

38
Consider the following electrical circuit
A
B
C
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?

39
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
40
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?

• Two maintenance calls for an old machine followed
  by a call for a new machine.
• Two maintenance calls for new machines followed
  by a call for an old machine.
• Three maintenance calls in a row for an old
  machine.
• Three maintenance calls in a row for a new machine

41
?????

• What is the probability that at least 2 people in
  this class (n39) have the same birthday Month
  and day?
• Year has 365 days forget leap year.
• Equally likelihood for each day

42
Multiplication of Choices

• If an operation can be performed in n1 ways, and
  if for each of these a second operation can be
  performed in n2 ways, and for each of the first
  two a third operation can be performed in n3
  ways, and so forth, then the sequence of k
  operations can be performed in n1 n2 nk
  ways.

43
Multiplication of Choices

• There are 5 processes needed to manufacture the
  side panel for a car clean, press, cut, paint,
  polish. Our plant has 6 cleaning stations, 3
  pressing stations, 8 cutting stations, 5 painting
  stations, and 8 polishing stations.
• How many different pathways through the
  manufacturing exist?
• What is the number of pathways that include a
  particular pressing station?
• What is the probability that a panel follows any
  particular path?
• What is the probability that a panel goes through
  pressing station 1?

44
Classical Definition of Probability
If an experiment can result in any one of N
different, but equally likely, outcomes, and if
exactly n of these outcomes corresponds to event
A, then the probability of event A is
45
Counting

• Suppose there are 3 vendors and we want to choose
  2. How many possible combinations of 2 can be
  chosen from the 3 vendors?

46
Counting - Combinations

• The number of combinations of n distinct objects
  taken r at a time is

n choose r
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Factorial Reminder

• n! n (n-1) (n-2) ..
• EX 4! (4)(3)(2)(1) 24
• 1! 1 0! 1

48
Counting

• 9 out of 100 computer chips are defective. We
  choose a random sample of n3.
• How many different samples of 3 are possible?
• How many of the samples of 3 contain exactly 1
  defective chip?
• What is the probability of choosing exactly 1
  defective chip in a random sample of 3?
• What is the probability of choosing at least 1
  defective chip in a random sample of 3?

49

• This class consists of 5 women and 34 men. If we
  randomly choose 4 people, what is the probability
  that there will be no women chosen?
• What is the probability that there will be at
  least 1 woman chosen?