1.8 Combinatorial Methods
41
examples in which more than one counting technique was required at different points
in the same problem. Sometimes, more than one technique is required to count the
elements of a single set.
Exercises
1. Two pollsters will can
90
Chapter 2 Conditional Probability
5. Suppose that on each play of a certain game, a person is
equally likely to win one dollar or lose one dollar. Suppose
also that the persons goal is to win two dollars by playing
this game. How large an initial fortu
2.2 Independent Events
75
Summary
A collection of events is independent if and only if learning that some of them occur
does not change the probabilities that any combination of the rest of them occurs.
Equivalently, a collection of events is independent
2.3 Bayes Theorem
2. Consider again the conditions of Example 2.3.4 in this
section, in which an item was selected at random from
a batch of manufactured items and was found to be defective. For which values of i (i = 1, 2, 3) is the posterior
probability
1.4 Set Theory
15
Summary
We will use set theory for the mathematical model of events. Outcomes of an experiment are elements of some sample space S, and each event is a subset of S. Two
events both occur if the outcome is in the intersection of the two s
1.5 The Definition of Probability
.
.
.
.
21
!
" #
If A1, . . . , Ak are disjoint, Pr ki=1Ai = ki=1 Pr(Ai ).
Pr(Ac ) = 1 Pr(A).
A B implies that Pr(A) Pr(B).
Pr(A B) = Pr(A) + Pr(B) Pr(A B).
It does not matter how the probabilities were determined. As lon
1.7 Counting Methods
25
Summary
A simple sample space is a finite sample space S such that every outcome in S has the
same probability. If there are n outcomes in a simple sample space S, then each one
must have probability 1/n. The probability of an even
50
Chapter 1 Introduction to Probability
It is shown in books on elementary calculus that the sum of the infinite series on
the right side of this equation is 1 (1/e), where e = 2.71828. . . . Hence, 1 (1/e) =
0.63212. . . . It follows that for a large va
32
Chapter 1 Introduction to Probability
parts are. For example, for i = 2, it does not require that the same n2 possibilities
be available for x2 regardless of what x1 is. It only requires that the number of
possibilities for x2 be n2 no matter what x1 i
2.1 The Definition of Conditional Probability
65
conditional on. For example, the multiplication rule for conditional probabilities becomes Pr(A1 A2 |B) = Pr(A1|B) Pr(A2 |A1 B). A partition is a collection of disjoint
events whose union is the whole sampl