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Chapter 1

Theory of Probability

1.1. Combinatorial Analysis

It is therefore necessary to be able to count the number of possible outcomes in various
common situations. In many cases the number of sample points in a sample space is not very
large and so direct enumeration or counting of sample points needed to obtain probabilities is
not difficult. However, problems arise where direct counting becomes a practical impossibility.In
such cases use is made of combinatorial analysis, which could be called a sophisticated way of
counting.

Fundamental principle of counting

If one thing can be accomplished in n 1 different ways and after this a second thing can be
accomplished in n 2 different ways, . . . , and finally a k-th thing can be accomplished in n k
different ways, then all k things can be accomplished in the specified order in n 1 · n 2 · · · n k
different ways.

Example 1.1. If a man has 2 shirts and 4 ties then he has 2·4 = 8 ways of choosing
a shirt and then a tie. ,

Permutations

Let us first consider a set of n objects that are all different. We may ask in how many ways these n
objects may be arranged, i.e. how many permutations of these objects exist. This is
straightforward to deduce, as follows: the object in the first position may be chosen in n different
ways, that in the second position in n − 1 ways, and so on until the final object is positioned. The
number of possible arrangements is therefore

n(n − 1)(n − 2) · . . . · 1 = n! (1.1)

Definition 1.1. Any ordered set which consists of n elements is called a
permutation and denoted by P n . It is read: permutation of n digits at a time. ✓

Permutations are computed by the formula

P n = n! (1.2)

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