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20 CHAPTER 2 PROBABILITY

depends on the objectives of the study. As specific questions occur later in the book, appro-
priate sample spaces are discussed.

EXAMPLE 2-2 If two connectors are selected and measured, the extension of the positive real line R is to take
the sample space to be the positive quadrant of the plane:

S R R
If the objective of the analysis is to consider only whether or not the parts conform to the
manufacturing specifications, either part may or may not conform. We abbreviate yes and no
as y and n. If the ordered pair yn indicates that the first connector conforms and the second
does not, the sample space can be represented by the four outcomes:

S 5yy, yn, ny, nn6

If we are only interested in the number of conforming parts in the sample, we might sum-
marize the sample space as

S 50, 1, 26
As another example, consider an experiment in which the thickness is measured until a
connector fails to meet the specifications. The sample space can be represented as

S 5n, yn, yyn, yyyn, yyyyn, and so forth6

In random experiments in which items are selected from a batch, we will indicate whether
or not a selected item is replaced before the next one is selected. For example, if the batch
consists of three items {a, b, c} and our experiment is to select two items without replace-
ment, the sample space can be represented as

S without 5ab, ac, ba, bc, ca, cb6
This description of the sample space maintains the order of the items selected so that the out-
come ab and ba are separate elements in the sample space. A sample space with less detail
only describes the two items selected {{a, b}, {a, c}, {b, c}}. This sample space is the possi-
ble subsets of two items. Sometimes the ordered outcomes are needed, but in other cases the
simpler, unordered sample space is sufficient.
If items are replaced before the next one is selected, the sampling is referred to as with
replacement. Then the possible ordered outcomes are

5aa, ab, ac, ba, bb, bc, ca, cb, cc6
S with
The unordered description of the sample space is {{a, a}, {a, b}, {a, c}, {b, b}, {b, c}, {c, c}}.
Sampling without replacement is more common for industrial applications.
Sometimes it is not necessary to specify the exact item selected, but only a property of the
item. For example, suppose that there are 5 defective parts and 95 good parts in a batch. To
study the quality of the batch, two are selected without replacement. Let g denote a good part
and d denote a defective part. It might be sufficient to describe the sample space (ordered) in
terms of quality of each part selected as

S 5gg, gd, dg, dd6