Mathematical Methods for Physicists: A concise introduction - Site Map

INTRODUCTION TO PROBABILITY THEORY
The numbers (14.5) are often called binomial coecients because they arise in
the binomial expansion
…x ‡ y† n ˆ x n ‡ n
x n1 y ‡ n
x n2 y 2 ‡‡
n
y n :
1
2
n
When n is very large a direct evaluation of n! is impractical. In such cases we use
Stirling's approximate formula
p
n!
2n n n e n :
The ratio of the left hand side to the right hand side approaches 1 as n !1. For
this reason the right hand side is often called an asymptotic expansion of the left
hand side.
Fundamental probability theorems
So far we have calculated probabilities by directly making use of the de®nitions; it
is doable but it is not always easy. Some important properties of probabilities will
help us to cut short our computation works. These important properties are often
described in the form of theorems. To present these important theorems, let us
consider an experiment, involving two events A and B, with N equally likely
outcomes and let
n 1 ˆ number of outcomes in which A occurs; but not B;
n 2 ˆ number of outcomes in which B occurs; but not A;
n 3 ˆ number of outcomes in which both A and B occur;
n 4 ˆ number of outcomes in which neither A nor B occurs:
This covers all possibilities, hence n 1 ‡ n 2 ‡ n 3 ‡ n 4 ˆ N:
The probabilities of A and B occurring are respectively given by
‡ n
P…A† 3
ˆn1
N ; P…B† ˆn2 ‡ n 3
N ; …14:6†
the probability of either A or B (or both) occurring is
‡ n
P…A ‡ B† 2 ‡ n 3
ˆn1
; …14:7†
N
and the probability of both A and B occurring successively is
P…AB† ˆn3
N :
Let us rewrite P…AB† as
P…AB† ˆn3
N ˆ n1 ‡ n 3 n 3
:
N n 1 ‡ n 3
…14:8†
486