Page 45 - Applied statistics and probability for engineers

Section 2-1/Sample Spaces and Events 23

Sample space S with events A and B A > B
A B A B

S S

(a) (b)

(A < B) > C (A > C)'
A B A B

A B

C S C S
S
(c) (d)

FIGURE 2-8 Venn diagrams. FIGURE 2-9 Mutually exclusive events.
The tree diagram in Fig. 2-6 describes the sample space of all possible vehicle types. The size
of the sample space equals the number of branches in the last level of the tree, and this quantity
equals 2 2 3 4× × × = 48. This leads to the following useful result.
Multiplication Rule
(for counting Assume an operation can be described as a sequence of k steps, and
techniques)
r the number of ways of completing step 1 is n 1 , and
r the number of ways of completing step 2 is n 2 for each way of completing step
1, and
r the number of ways of completing step 3 is n 3 for each way of completing step 2,
and so forth.
The total number of ways of completing the operation is
n 1 × n 2 × … × n k

Example 2-9 Web Site Design The design for a Website is to consist of four colors, three fonts, and three
positions for an image. From the multiplication rule, 4 3 3× × = 36 different designs are possible.
Practical Interpretation: The use of the multipication rule and other counting techniques enables one to easily deter-
mine the number of outcomes in a sample space or event and this, in turn, allows probabilities of events to be determined.

Permutations

Another useful calculation inds the number of ordered sequences of the elements of a
,
,
set. Consider a set of elements, such as S = { a b c}. A permutation of the elements is an
ordered sequence of the elements. For example, abc acb bac bca cab, and cba are all of
,
,
,
,
the permutations of the elements of S.
The number of permutations of n different elements is n! where
n! = ×( n − ) ×( n − ) ×2 … × ×1 (2-1)
1
2
n