Page 39 - Probability and Statistical Inference

16 1. Nations of Probability

1.4.3 Selected Counting Rules
In many situations, the sample space S consists of only a finite number of
equally likely outcomes and the associated Borel sigma-field ß is the collection
of all possible subsets of S, including the empty set ϕ and the
whole set S. Then, in order to find the probability of an event A, it will be
important to enumerate all the possible outcomes included in the sample space
S and the event A. This section reviews briefly some of the customary count-
ing rules followed by a few examples.
The Fundamental Rule of Counting: Suppose that there are k different
th
tasks where the i task can be completed in n ways, i = 1, ..., k. Then, the total
i
number of ways these k tasks can be completed is given by .
Permutations: The word permutation refers to arrangements of some ob-
jects taken from a collection of distinct objects.

The order in which the objects are laid out is important here.

Suppose that we have n distinct objects. The number of ways we can arrange k
of these objects, denoted by the symbol P , is given by
n
k

n
The number of ways we can arrange all n objects is given by P which is
n
denoted by the special symbol

We describe the symbol n! as the “n factorial”. We use the convention to inter-
pret 0! = 1.
Combinations: The word combination refers to the selection of some ob-
jects from a set of distinct objects without regard to the order.

The order in which the objects are laid out is not important here.

Suppose that we have n distinct objects. The number of ways we can select k of
these objects, denoted by the symbol , is given by

But, observe that we can write n! = n(n – 1)...(n – k + 1)(n – k)! and hence we
can rewrite (1.4.10) as