Chapter 4 - Introduction to Probability p.173

Question 1

Addition law p.191

Answer 1

A probability law used to compute the probability of the union of two events. It is
Probability(Union(A,B)) = Probability(A) + Probability(B) - Probability(Intersect(A,B))

For mutually exclusive events, Probability(Intersect(A,B)) = 0; in this case the addition law reduces to Probability(Union(A,B)) = Probability(A) + Probability(B)

Question 2

Basic requirements for assigning probabilities p.180

Answer 2

§ Two requirements that restrict the manner in which the probability assignments can be made:

    1. For each experimental outcome E we must have 0 <= P€ <= 1;
    2. Considering all experimental outcomes, we must have P(E1) + P(E2) + . . . + P(En) = 1.0

Question 3

Bayes’ theorem p.204

Answer 3

A method used to compute posterior probabilities.

Question 4

Classical method p.180

Answer 4

A method of assigning probabilities that is appropriate when all the experimental outcomes are equally likely.

Question 5

Combination p.179

Answer 5

In an experiment we may be interest in determining the number of ways n objects may be selected from among N objects without regard to the order in which the n objects are selected. Each selection of n objects is called a combination and the total number of combinations of N objects taken n at a time is

Combination(N,n) = (N . n) = N!/(n!(N - n)!) for n = 0,1,2,…,N.

Question 6

Complement of A p.189

Answer 6

The event consisting of all sample points that are not in A

Question 7

Conditional probability p.196

Answer 7

The probability of an event given that another event already occurred. The conditional probability of a given B is
P(A | B) = P(Intersect(A,B))/P(B)

Question 8

Event

Answer 8

A collection of sample points.

Question 9

Experiment

Answer 9

A process that generates well-defined outcomes.

Question 10

Independent events p.199

Answer 10

Two events A and B where P(A|B) = P(A) or P(B|A) = P(B); that is, the events have no influence on each other.

Question 11

Intersection of A and B p.191

Answer 11

The event containing the sample points belonging to both A and B.
The intersection is denoted Intersect(A,B)

Question 12

Joint probability p.197

Answer 12

The probability of two events both occurring; that is, the probability of the intersection of two events.

Question 13

Marginal probability p.197

Answer 13

The values in the margins of a joint probability table that provide the probabilities of each event separately.

Question 14

Multiple-step experiment p.176

Answer 14

An experiment that can be described as a sequence of steps. If a multiple-step experiment has k steps with n1 possible outcomes on the first step, n2 possible outcomes on the second step, and so on, the total number of experiment outcomes is given by (n1)(n2)…(nk)

Question 15

Multiplication law p.199

Answer 15

A probability law used to compute the probability of the intersection of two events. It is
P(Intersect(A,B)) = P(B)P(A|B), or
P(Intersect(A,B)) = P(A)P(B|A).

For independent events it reduces to
P(Intersect(A,B)) = 0.

Question 16

Mutually exclusive events p.193

Answer 16

Events that have no sample points in common; that is, Intersect(A,B) is empty and Probability(Intersect(A,B)) = 0.

Question 17

Permutation p.179

Answer 17

In an experiment we may be interested in determining the number of ways n objects may be selected from among N objects when the order in which the n objects are selected is important. Each ordering of n objects is called a permutation and the total number of permutations of N objects taken n at a time is

Permutations(N,n) = n!(N . n) = N!/(N - n)! For n = 0, 1, 2, …, N.

Question 18

Posterior probabilities p.205

Answer 18

Revised probabilities of events based on additional information.

Question 19

Prior probabilities p.204

Answer 19

Initial estimates of the probabilities of events.

Question 20

Probability p174

Answer 20

A numerical measure of the likelihood that an event will occur.

Question 21

Relative frequency method p.181

Answer 21

A method of assigning probabilities that is appropriate when data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times.

Question 22

Sample point p.176

Answer 22

An element of the sample space. A sample point represents an experimental outcome.

Question 23

Sample space p.176

Answer 23

The set of all experimental outcomes.

S = {Head, Tail}

Question 24

Subjective method p.181

Answer 24

A method of assigning probabilities on the basis of judgement.

Question 25

Tree diagram p.177

Answer 25

A graphical representation that helps in visualizing a multiple-step experiment.

Question 26

Union of A and B p.190

Answer 26

The event containing all samples points belonging to A or B or both.

The union is denoted: Union(A,B)

Question 27

Venn diagram p.189

Answer 27

A graphical representation for showing symbolically the sample space and operations involving events in which the sample space is represented by a rectangle and events are represented as circles within the sample space.

Question 28

4.1 Counting Rule for Combinations

Answer 28

The number of combinations of N objects taken n at a time is
Combination(N,n) =(N . n) = N!/(n!(N - n)!)
Where
N! = N(N - 1)(N - 2) . . . (2)(1)
n! = n(n- 1)(n - 2) . . . (2)(1)
And, by definition,
0! = 1

Question 29

4.2 Counting Rule for Permutations

Answer 29

The number of permutations of N objects taken nat a time is given by

PNn = n!(N . n) = N!/(N - n)! , For n = 0, 1, 2, …, N.

Question 30

4.5 Computing Probability Using the Complement

Answer 30

Probability(A) = 1 - Probability(Complement(A))

Question 31

4.6 Addition Law

Answer 31

Probability(Union(A,B)) = Probability(A) + Probability(B) - Probability(Intersect(A,B))

Question 32

4.7 Conditional Probability

Answer 32

P(A|B) = P(Intersect(A,B))/P(B)
P(B|A) = P(Intersect(A,B))/P(A)

Question 33

4.11 Multiplication Law

Answer 33

P(Intersect(A,B)) = P(B)P(A|B)
P(Intersect(A,B)) = P(A)P(B|A)

Question 34

4.13 Multiplication Law for Independent Events

Answer 34

P(Intersect(A,B)) = P(A)P(B)

Question 35

4.19 Bayes’ Theorem

Answer 35

P(Ai|B) = P(Ai)P(B|Ai) / [P(A1)P(B|A1) + P(A2)P(B|A2) + . . . P(An)P(B|An)]

Question 36

Random experiment

Answer 36

A random experiment is a process that generates well-defined experimental outcomes. On any single repetition or trial, the outcome that occurs is determined completely by chance.
n! = n(n- 1)(n - 2) . . . (2)(1)
1. The experimental outcomes are well defined, and in many cases can even be listed prior to conducting the experiment.
2. On any single repetition or trial of the experiment, one and only one of the possible experimental outcomes will occur.
3. The experimental outcome that occurs on any trial is determined solely by chance.

Question 37

Samples space

Answer 37

The sample space for a random experiment is the set of all experimental outcomes.

Question 38

Counting rule for multiple-step experiments

Answer 38

If an experiment can be described as a sequence of k steps with n1 possible outcomes on the first step, n2 possible outcomes on the second step, and so on, the total number of experiment outcomes is given by (n1)(n2)…(nk)

Question 39

Basic Requirements for Assigning Probabilities

Answer 39

1. The probability assigned to each experimental outcome must be between 0 and 1, inclusively. If we let Exp(i) denote the ith experimental outcome and Probability(Exp(i)) its probability, then this requirement can be written as
   • 0 <= Probability(Exp(i)) <= 1 for all I
2. The sum of the probabilities for all the experimental outcomes must equal 1.0. Fr n experimental outcomes, this requirement can be written as
   • Probability(Exp(1)) + Probability(Exp(2)) + . . . + Probability(Exp(n)) = 1

Question 40

Event

Answer 40

An event is a collection of sample points.

Question 41

Probability of an event

Answer 41

The probability of any event is equal to the sum of the probabilities of the sample points in the event.

The probability of event C is given by:
Probability(c) = P(2, 6) + P(2, 7) + P(2, 8) + P(3, 6) + P(3, 7)

Question 42

Union of two events

Answer 42

The union of A and B is the event containing all sample points belonging to A or B or both.

The union is denoted by Union(A,B)

Question 43

Intersection of Two Events

Answer 43

Given two events A and B, the intersection of A and B is the event containing the sample points belonging to both A and B.

The intersection is denoted Intersect(A,B)

Question 44

Mutually Exclusive Events

Answer 44

Two events are said to be mutually exclusive if the events have no sample points in common.

Question 45

Addition Law for Mutually Exclusive Events

Answer 45

P(Union(A,B)) = P(A) + P(B)

Question 46

Independent events

Answer 46

Two events A and B are independent if
P(A|B) = P(A), or
P(B|A) = P(B)
Otherwise, the events are dependent.

Question 47

Bayes’ Theorem (Two-Event Case)

Answer 47

P(A1|B) = P(A1)P(B|A2) / [P(A1)P(B|A1) + P(A2)P(B|A2)]

P(A2|B) = P(A2)P(B|A2) / [P(A1)P(B|A1) + P(A2)P(B|A2)]