Applied Statistics and Probability for Engineers

2-1.4 Counting Techniques (CD Only)

In many of the examples in Chapter 2, it is easy to determine the number of outcomes in each

event. In more complicated examples, determining the number of outcomes that comprise the

sample space (or an event) becomes more difficult. To associate probabilities with events, it is

important to know the number of outcomes both in an event and in the sample space. Some

simple rules can be used to simplify the calculations.

In Example 2-4, an automobile manufacturer provides vehicles equipped with selected

options. Each vehicle is ordered

With or without an automatic transmission

With or without air conditioning

With one of three choices of a stereo system

With one of four exterior colors

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.

2-1

If an operation can be described as a sequence of ksteps, and

if the number of ways of completing step 1 is n 1 , and

if the number of ways of completing step 2 is n 2 for each way of completing

step 1, and

if 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 pnk

Multiplication

Rule (for

counting

techniques)

EXAMPLE S2-1 In the design of a casing for a gear housing, we can use four different types of fasteners,

three different bolt lengths, and three different bolt locations. From the multiplication rule,

4  3  3 36 different designs are possible.

Permutations

Another useful calculation is the number of ordered sequences of the elements of a set.

Consider a set of elements, such as S{a, b, c}. A permutationof the elements is an ordered

sequence of the elements. For example, abc, acb, bac, bca, cab, and cbaare all of the permu-

tations of the elements of S.

The number of permutationsof ndifferent elements is where

n!n 1 n 12  1 n 22 p 2  1 (S2-1)

n!

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