Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.3 PERMUTATIONS AND COMBINATIONS

We note that (30.27) may be written in a more general form ifSis not simply

divided intoAandA ̄but, rather, intoanyset of mutually exclusive eventsAithat

exhaustS. Using the total probability law (30.24), we may then write

Pr(B)=

∑

i

Pr(Ai)Pr(B|Ai),

so that Bayes’ theorem takes the form

Pr(A|B)=

Pr(A)Pr(B|A)

∑

iPr(Ai)Pr(B|Ai)

, (30.28)

where the eventAneed not coincide with any of theAi.

As a final point, we comment that sometimes we are concerned only with the

relativeprobabilities of two eventsAandC(say), given the occurrence of some

other eventB. From (30.26) we then obtain a different form of Bayes’ theorem,

Pr(A|B)

Pr(C|B)

=

Pr(A)Pr(B|A)

Pr(C)Pr(B|C)

, (30.29)

which does not contain Pr(B) at all.

30.3 Permutations and combinations

In equation (30.5) we defined the probability of an eventAin a sample spaceSas

Pr(A)=

nA

nS

,

wherenAis the number of outcomes belonging to eventAandnSis the total

number of possible outcomes. It is therefore necessary to be able to count the

number of possible outcomes in various common situations.

30.3.1 Permutations

Let us first consider a set ofnobjects that are all different. We may ask in

how many ways thesenobjects may be arranged, i.e. how manypermutationsof

these objects exist. This is straightforward to deduce, as follows: the object in the

first position may be chosen inndifferent 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! (30.30)

Generalising (30.30) slightly, let us suppose we choose onlyk(<n) objects

fromn. The number of possible permutations of thesekobjects selected fromn

is given by

n(n−1)(n−2)···(n−k+1)

︸ ︷︷ ︸

kfactors

=

n!

(n−k)!

≡nPk. (30.31)