Probability

1
Chapter 2

• Probability

2
LEARNING OBJECTIVES

• Understand and describe sample spaces and events
• Interpret probabilities and use probabilities of
  outcomes to calculate probabilities of events in
  discrete sample spaces
• Calculate the probabilities of joint events such
  as unions and intersections from the
  probabilities of individual events
• Interpret and calculate conditional probabilities
  of events
• Determine the independence of events and use
  independence to calculate probabilities
• Use Bayes theorem to calculate conditional
  probabilities

3
Sample Space and Events

• An experiment is any action or process that
  generates observations
• The sample space of an experiment, denoted S, is
  the set of all possible outcomes, or sample
  points.
• Example Toss a fair coin 3 times in a row
• The sample space has 8 sample points.
• SHHH, THH, HHT, THT, HTH, TTH, HTT, TTT

• An event is a subset of the sample space S
• Example look at 3 different events of previous
  example
• The event of 3 heads,
• A HHH
• The event of 2 heads,
• BHHT, HTH, THH
• The event that the last toss is a head,
• CHHH, HTH, THH, TTH

4
Set Relations

• Suppose S is the universal set, with two subsets,
  A and B
• A set, A, is a subset of B if all elements of A
  belong to B, A?B

• The union of two events A and B, denoted by A?B,
  and read A or B, is the event consisting of all
  elements that are either in A, in B, or in both
• Or the union A?B x x ? A or x ? B

S
B
A
S
5
Set Relations-Cont.

• The intersection of two events A and B, denoted
  by AnB, and read A and B, is the event
  consisting of all elements that are in both
• The intersection AnBx x ?A and x ?B is the
  subset of S which contains all elements that are
  in both A B

• The complement of an event A, denoted by A, is
  the set of all elements in S that are not
  contained in A

S A
A
S
A
6
Set Relations-Cont.

• Sets A and B are mutually exclusive or disjoint,
  if and only if AnB Ø, or events A B have no
  elements in common

• Any number of sets, A1, A2, A3, are mutually
  exclusive if and only if AinAj Ø for i?j.
• A1 nA2 Ø A2 n A3 Ø
• A1 n A3 Ø A2 n A4 Ø
• A1 n A4 Ø A3 n A4 Ø

S
S
A1
A2
A3
A4
7
Example1

• For an experiment, let
• A 0,1,2,3,4,
• B 3,4,5,6, and
• C 1,3,5
• Determine
• A?B
• A?C
• A?B
• A?C
• A
• A?C

• Solution
• A?B0,1,2,3,4,5,6
• A?C0,1,2,3,4,5
• A?B3,4
• A?C1,3
• A5,6
• A?C6

8
Example2

• The rise time of a reactor is measured in minutes
  (and fractions of times). Let the sample space
  positive, real numbers. Define the events A and B
  as follows
• Ax xlt72.5 and Bx xgt52.5
• Describe each of the following events.
• a) A
• b) B
• c) A?B
• d) A?B

• Solution
• a) A? x x ? 72.5
• b) B? x x ? 52.5
• c) A ? B x 52.5 lt x lt 72.5
• d) A ? B x x gt 0

9
Example3

• In an injection-molding operation, several
  characteristics of each molded part are evaluated
• Let A denote the event that a part meets customer
  shrinkage requirements, B denote the event that a
  part meets customer color requirements, and C
  denote the event that a critical length meets
  customer requirements
• a) Construct a Venn diagram that includes these
  events and indicate the region in the diagram in
  which a part meets all customer requirements.
  Shade the areas that represent the following
• b) B?C
• c) A?B
• d) A?B

• Solution

10
Class Problem

• Disks of polycarbonate plastic from a supplier
  are analyzed for scratch resistance and shock
  resistance. The results from 100 disks are
  summarized below.
• Shock Resistance
• High low
• Scratch High 70 9
• Resistance low 16 5
• Let A denote the event that a disk has high shock
  resistance, and let B denote the event that a
  disk has high scratch resistance. Determine the
  number of disks in AnB, A, and A?B

• Solution
• Number of samples in AnB
• Number of samples in A'
• Number of samples in A?B

11
Interpreting Probabilities

• The assignment of a weight between 0 and 1 to
  indicate the likelihood of the occurrence of an
  event.
• The probability of an event is defined in terms
  of an experiment and a sample space.

• Example Toss a fair coin three times in a row
• The probability of getting 3 heads P(A) 1/8
• The probability of getting 2 heads P(B)3/8
• The probability that the last toss is a head P(C)
  4/8 1/2

12
Axioms of Probability

• For any event A, P (A) ?0
• The chance of occurring should be at least 0
• P(S)1
• The maximum possible probability is assigned to S
• Let A1, A2, A3,, An, be a finite or infinite
  sequence of mutually exclusive events. Then
• P(A1?A2?A3) P (A1) P(A2) P(A3)? P(Ai)

• Example
• If an experiment has the three possible and
  mutually exclusive outcomes A, B, and C, check in
  each case whether the assignment of probabilities
  is permissible
• P(A) 1/3, P(B) 1/3, and P(C) 1/3
• P(A) 0.64, P(B) 0.38, and P(C) -.02
• P(A) 0.35, P(B)0.52, and P(C) 0.26
• P(A) 0.57, P(B)0.24, and P(C) 0.19

13
Class Problem

• The sample space of a random experiment is a, b,
  c, d, e with probabilities 0.1, 0.1, 0.2, 0.4,
  and 0.2, respectively. Let A denote the event a,
  b, c, and let B denote the event c, d, e.
  Determine the following
• a) P(A)
• b) P(B)
• c) P(A')
• d) P(A?B)
• e) P(AnB)

• Solution
• a) P(A)
• b) P(B)
• c) P(A')
• d) P(A?B)
• e) P(A?B)

14
Rules of Probability Complement Rule

• The probability of impossible events is 0 P(Ø)
  0
• Complement rule P(A) 1- P(A)

• Proof
• From axioms 3 for finite case, let k2, A1A and
  A2A'
• By definition, A?A' S while A and A' are
  mutually exclusive
• 1P(S)P(A?A')P(A)P(A' )
• P(A) 1- P(A)

S A
A
15
Addition Rule

• For any two events A and B
• P(A1?A2)P(A1)P(A2) P(A1nA2)
• By Venn diagram

• The probability of a union of more than two
  events
• P(A1?A2?A3)
• P(A1) P(A2) P(A3)
• P(A1nA2nA3)
• -P(A1nA2) -P(A2nA3) -P(A1nA3)

16
Example 1

• If P(A) 0.3, P(B)0.2, and P(A?B) 0.1,
  determine the following probabilities
• a) P(A)
• b) P(A?B)
• c) P(A?B)
• d) P(A?B)
• e) P(A?B)
• f) P(A?B)

• Solution
• a) P(A') 1- P(A) 0.7
• b) P(A?B) P(A) P(B) - P(A?B) 0.30.2 - 0.1
  0.4
• c) P(A?B) P(A?B) P(B). Therefore, P(A?B)
  0.2 - 0.1 0.1
• d) P(A) P(A?B) P(A?B) Therefore, P(A?B)
  0.3 - 0.1 0.2
• e) P((A?B) ') 1 - P(A?B) 1 - 0.4 0.6
• f) P(A?B) P(A') P(B) - P(A ? B) 0.7 0.2 -
  0.1 0.8 from part c

17
Example2

• Denote the six events 1,2,3,4,5, and 6 associated
  with tossing a six-sided die once by E1, E2, E3,
  E4, E5, and E6
• Suppose the die is constructed so that any of the
  three even outcomes is twice as likely to occur
  as any of the three odd outcomes
• Determine P(A) where the event A is even
• Determine P(B) where the event B is less than or
  equal to 3

• Solution
• P(E1)P(E3)P(E5)
• P(E2)P(E4)P(E6)
• Define Aoutcome is even E2 ? E4 ? E6
• P(A) P(E2)P(E4)P(E6)
• Boutcome 3 E1?E2?E3
• P(B) P(E1)P(E2)P(E3)

18
Class Problem

• Disks of polycarbonate plastic from a supplier
  are analyzed for scratch resistance and shock
  resistance. The results from 100 disks are
  summarized below.
• Shock Resistance
• High low
• Scratch High 70 9
• Resistance low 16 5
• If a disk is selected at random, what is the
  probability that its scratch resistance is high
  and its shock resistance is high?
• b) If a disk is selected at random, what is the
  probability that its scratch resistance is high
  or its shock resistance is high?
• Consider the event that a disk has high scratch
  resistance and the event that a disk has high
  shock resistance. Are these two events mutually
  exclusive?

• Solution
• Let A denote the event that a sample has high
  shock resistance and let B denote the event that
  a sample has high scratch resistance.
• a) P(A?B)
• b) P(A?B) P(A) P(B) - P(A?B)
• c) Because (A?B) does not equal Ø , A and B

19
Equally Likely Outcomes

• In an experiment consisting of N outcomes, it is
  reasonable to assign equal probabilities to all N
  sample events
• So, p1/N

• Example
• When two dice are rolled separately, there are
  N36 outcomes, which are equally likely or
  P(Ei)1/36
• Let Asum of two numbers7
• P(A)

20
Counting Techniques

• Ability to count number of elements in the sample
  space without listing actually each element
• The Product Rule for Ordered Pairs
• If the first object of an ordered pair can be
  selected in n1 ways, and for each of these n1
  ways the second object of the pair can be
  selected in n2 ways, then the number of pairs is
  n1 n2

• Example
• A homeowner requires two types of contractors,
  plumbing and electrical
• 3 plumbing contractors
• 3 electrical contractors
• How many possible ways of choosing the two types
  of contractors?
• N n1 n2 339

21
Tree Diagrams

• Used to represent pictorially all the
  possibilities
• Starting on the left side of the diagram, for
  each possible first element of a pair a
  straight-line segment emanates rightward
• Construct another line segment emanating from the
  tip of the branch for each possible choice of a
  second element of the pair
• A more general Product Rule
• N n1 n2 n3 nk

• Example

E1
E2
E3
P1
E1
P2
E2
P3
E3
E1
E2
E3
22
Permutations

• Any ordered sequence of k objects taken from a
  set of n distinct objects is called a permutation
  of size k of the objects
• The number of permutations of size k that can be
  formed from the n objects is denoted by Pk,n
• Obtained from the general product rule
• Pk,nn(n-1)(n-2)(n-k2)(n-k1)
• Using factorial notation

• Example
• Consider the set A,B,C,D,E consisting of 5
  elements
• Number of permutations of size 3?
• By taking 5 letters three at a time
• P5,3 5!/(5-3)! 60
• Class Problem
• Three awards will be given for a class of 25
  graduate students
• If each student can receive at most one award
• How many possible selections?
• P25,3

23
Combinations

• Example
• Consider the set A,B,C,D,E consisting of 5
  elements
• Number of combinations of size 3?
• There are six permutations of size 3 consisting
  of the elements A, B, and C (3!6)
• These six permutations are equivalent to the
  single combination A,B,C
• So, 60/3! 10
• In general

• Any unordered sequence of k objects taken from a
  set of n distinct objects is called combination
  of size k of the objects
• The number of combinations of size k that can be
  formed from n distinct objects will be denoted by
  
• Smaller than the number of permutations because
  the order is disregarded

24
Class Problem

• Car dealer service 10 foreign cars and 15
  domestic cars on a particular day
• There are only 6 mechanics
• If 6 cars are chosen at random, what is the
  probability that exactly 3 of the cars are
  domestic and the other cars are foreign?
• What is the probability that at least 3 of the
  cars are domestic and the other cars are foreign?

• Solution
• Let D3exactly 3 of the 6 cars chosen are
  domestic
• P(D3)
• P(D3?D4 ?D5 D6)

25
Conditional Probability

• Interested at the probability of an event
  occurring conditional on the knowledge that
  another event has occurred
• Let A and B be events with P(A) ?0
• The conditional probability of B given A is
• P(BA) P(AnB)/P(A)
• Note P(BA) is undefined if P(A) 0.

• Simple example concerns the situation in which
  two events are mutually exclusive
• P(BA) P(AnB)/P(A)0/P(B)0

A
B
26
Examples

• If the probability that a research project will
  be planned is 0.80 and the probability that it
  will be planned and well executed is 0.72, what
  is the probability that a research project, which
  is well planned, will also be well executed?
• Solution
• P(A/B) P(AnB)/P(B)0.72/0.800.90

• If the probability that a communication system
  will have high fidelity is 0.81, and the
  probability that it will have high fidelity and
  high selectivity is 0.18, what is the probability
  that a system with high fidelity will also have
  high selectivity?
• A A communication system which has high
  selectivity
• B A communication system which has high fidelity
• Data P(B)0.81, P(AnB)0.18
• P(A/B)P(AnB)/P(B)
• 0.18/0.812/9

27
Class Problem

• A magazine publishes three columns entitled A, B,
  and C
• Reading habits of a selected readers with respect
  to these columns
• Read regularly A B C AnB AnC BnC AnBnC
• Probability 0.14 0.23 0.37 0.08 0.09 0.13 0.05
• P(AB)
• P(AB ? C)
• P(A reads at least one) P(AA?B?C)
• P(A?B C)

28
Class Problem

• Disks of polycarbonate plastic from a supplier
  are analyzed for scratch resistance and shock
  resistance. The results from 100 disks are
  summarized below.
• Shock Resistance
• High low
• Scratch High 70 9
• Resistance low 16 5
• Let A denote the event that a disk has high shock
  resistance, and let B denote the event that a
  disk has high scratch resistance. Determine the
  following probabilities.
• P(A)
• P(B)
• P(A/B)
• P(B/A)

• Solution
• P(A)
• P(B)
• P(A/B)
• P(B/A)

29
Multiplication Rule

• From the definition of conditional probability,
  we can write
• P(AnB) P(A) P(B/A)
• Example The supervisor of a construction group
  (20 workers) wants to get the opinion of 2 of
  them about new safety regulations. If 12 of them
  favor the new regulations and the other 8 are
  against it, what is the probability that both of
  them chosen by the supervisor will be against the
  new safety regulation?

• Solution
• P(A)the first worker selected will be against
  the new safety regulation8/20
• P(B/A)the second worker selected will be against
  given that the first one is against7/19
• P(AnB) P(A) P(B/A)
• (8/20)(7/19)14/95

30
Independent Events

• Two events A and B are independent if and only if
• P(B/A)P(B) if P(A)?0 and
• P(A/B)P(A) if P(B)?0
• P(B/A)P(B) means the probability of event B
  remains the same whether or not event A is
  conditional upon
• One event does not affect the probability of the
  another event
• P(AnB)P(A)P(B/A) P(A)P(B)

• Example 1
• What is the probability of getting two heads in
  two flips of a balanced coin?
• Solution
• (1/2)(1/2)1/4
• Example 2
• If P(C)0.65, P(D)0.4, and P(CnD)0.24, are the
  events C and D independent?
• Solution
• P(C)P(D)(0.65)(0.4)0.26?0.24

31
Class Problem

• Solution
• P(AnB)
• P(A)
• P(B) 79/100

• Disks of polycarbonate plastic from a supplier
  are analyzed for scratch resistance and shock
  resistance. The results from 100 disks are
  summarized below.
• Shock Resistance
• High low
• Scratch High 70 9
• Resistance low 16 5
• Let A denote the event that a disk has high shock
  resistance, and let B denote the event that a
  disk has scratch resistance. Are events A and B
  independent?

32
Total Probability Rule

• Let A1, A2, A3,, An be a collection of mutually
  exclusive events with known probabilities which
  partition S
• Consider an event B with known conditional
  probabilities
• How to use P(Ai) and P(B/Ai) to calculate P(B)
• Easily achieved by
• B(A1nB)? ?(AnnB)

• Events AinB are mutually exclusive
• P(B)P(A1nB) P(AnnB)
• P(A1)P(B/A1)P(An)P(B/An)
• Known as the law of total probability
• Example
• Suppose that P(A/B)0.2, P(A/B')0.3, and
  P(B)0.8. What is P(A)?
• Solution

A1
A2
An
B
33
Example

• In a certain assembly plant, three machines, B1,
  B2, and B3, make 30, 45, and 25, respectively,
  of the products
• It is known from past experience that 2, 3, and
  2 of the products made by each machine,
  respectively, are defective
• Select a finished product randomly
• What is the probability that it is defective?

• Let
• A the product is defective
• B1the product is made by machine B1
• B2the product is made by machine B2
• B3the product is made by machine B3
• P(A)P(B1)P(A/B1)P(Bn)P(A/Bn)
• P(B1)P(A/B1)(0.3)(0.02)0.006
• P(B2)P(A/B2)(0.45)(0.03)0.0135
• P(B3)P(A/B3)(0.25)(0.02)0.005
• P(A)0.0060.01350.0050.0245

34
Posterior Probabilities

• How to use the probabilities P(Ai) and P(B/Ai) to
  calculate the probabilities P(Ai/B)
• This is the revised probabilities of the events
  Ai conditional on the event B
• Assume P(A1), , P(An) are the prior
  probabilities of events A1, , An
• Observation of the event B provides some
  additional information to revise these prior
  probabilities
• Called posterior probabilities and conditional on
  the event B

• From the law of total probability
• Known as Bayes Theorem

35
Example

• Let
• A the product is defective
• B1the product is made by machine B1
• B2the product is made by machine B2
• B3the product is made by machine B3
• Instead of asking P(A), we want to find P(Bi/A)
• Using the Bayes formula
• P(B3/A)0.005/0.0245 10/49

• In a certain assembly plant, three machines, B1,
  B2, and B3, make 30, 45, and 25, respectively,
  of the products
• It is known from past experience that 2, 3, and
  2 of the products made by each machine,
  respectively, are defective
• Suppose a product is selected randomly and it is
  defective
• What is the probability made by machine Bi (say
  B3)?

36
Prior and Posterior Probabilities

• Prior probabilities
• It is known from past experience that 2, 3, and
  2 of the products made by each machine,
  respectively, are defective
• Past data
• B1the product is made by machine B1
• B2the product is made by machine B2
• B3the product is made by machine B3
• P(B1)0.02
• P(B2)0.03
• P(B3)0.02

• Posterior probabilities
• Revision of the prior probabilities conditional
  on the event A
• A product is selected randomly and it is
  defective, what is the probability made by
  machine Bi?
• P(B1/A)0.006/0.0245
• P(B2/A)0.0135/0.0245
• P(B3/A)0.005/0.0245 10/49

37
Class Problem

• A TV store sells 3 different brands of TVs
• Of its TV sales, 50 are brand 1, 30 are brand
  2, and 20 are brand 3
• Known that 25 of brand 1 requires warranty
  repair work, whereas brand 2 and 3 are 20 and
  10, respectively
• What is the probability that a selected buyer has
  a TV that will need repair while under warranty?
• If a customer returns a TV brand, what is the
  probability that is a brand 1? A brand 2? A brand
  3?

38
Solution
39
Tree Diagram
P(B/A1) P(A1)P(B?A1)0.125
P(B/A1)0.25
Repair
No repair
P(B'/A1)0.75
P(A1)0.5
P(B/A2) P(A2)P(B?A2)0.06
Brand 1
Repair
P(B/A2)0.2
P(A2)0.3
Brand 2
No repair
P(B'/A2)0.8
P(A3)0.3
Brand 3
P(B/A3) P(A3)P(B?A3)0.02
Repair
P(B/A3)0.1
No repair
P(B'/A3)0.90
P(B)P(brand1 and repair) or (brand 2 and
repair) or (brand 3 and repair) 0.205 P(B)
P(B?A1) P(B?A2) P(B?A3)0.1250.060.02 0.205
P(A1/B) P(A2/B) P(A3/B)
40
Class Problem

• An assembly plant receives its voltage regulators
  from three different suppliers, 60 from supplier
  B1, 30 from supplier B2, and 10 from supplier
  B3. If 95 of the voltage regulators from B1, 80
  of those from B2, and 65 of those from B3
  perform according to specifications, what is the
  probability that a particular voltage regulator
  which is known to perform according to
  specifications came from supplier B3?

• If A denotes the event a voltage regulator
  received by the plant performs according to
  specifications and B1, B2, B3 are the events that
  it comes from the respective suppliers, then