Mathematics and Probability
http://www.maths.lth.se/matstat/staff/bengtr
Basics
In probability theory, or rather probability calculus, one starts with
given probabilities and computes new ones. As long as one agrees to
measure probability on an increasing linear scale between 0 and 1, the
following rules are obvious.
 Probabilities are numbers between 0 and 1, endpoints included.
 If an event surely occurs, its probability equals 1.
 If an event can be decomposed into smaller events, its probability
equals the sum of the probabilities of the subevents.
As a consequence, the probability that an event does not occur equals one
minus the probability that it does occur.
 For instance,
P(X > 2) = 1 − P(X ≤ 2).
 To compute the probability that two events occur simultaneously, one
multiplies the probability of one of them with the conditional
probability of the other one.
 Example: An urn contains 5 white and
3 black balls, and one draws two balls without replacement. The probability
that both are white is given by
P(A and B) = P(A)⋅P(BA)= 
5 5+3 
⋅ 
4 4+3 
= 
20 56 
= 
5 14 
where A denotes "first ball white" and B denotes "second ball white", and
the  sign denotes "given" or "conditionally on".
The values of P(A) and
P(BA) might have been obtained from the previous rule and symmetry; at each
step all the remaining balls have the same probability to be drawn. Note also,
to avoid potential misunderstandings, that P(B) = 5/8.
The above means that probabilities are treated axiomatically ;
for instance, if one claims
P(rain tomorrow) = 0.7,
then, whether this means that 70% of the days are rainy ones, or that 70% of
the days of this time of the year are rainy, or some meteorological method
has been used, or that it is merely based on a pessimistic feeling, one has
to admit that
P(not rain tomorrow) = 0.3,
since, otherwise, one contradicts oneself. In probability theory the above
reasonings are formalised into rules which guarantee consistent, i. e.
contradictionfree, results.
Relation to measure
The events considered are of the type:
The outcome of some observed random phenomenon falls within a given set.

For some events one is faced with given probabilities according to
P(outcome ∈ A) = μ(A)
where A is a subset of all possible outcomes of the phenomenon in question and
μ is a suitable function, which is thought of as assigning
probability mass to the subsets.
For instance:

In elementary courses one has
μ(A) = ∑_{k ∈ A} p_k,
with p_k ≥ 0 and ∑_{k} p_k = 1,
if the outcome is an integer, and
μ(A) = ∫_{x ∈ A} f(x)dx,
with f(x) ≥ 0 and ∫_{x} f(x) dx = 1,
if the outcome is a real number. If the outcome is a pair of two numbers, one
has a double sum and p_{jk} or a double integral and f(x, y).
In exercises, one is given the p or the f and is required to compute some
interesting probability.
Sometimes no μ is given. Instead one has to use symmetry, for instance
when a die is thrown or when one counts the number of heads when tossing a
coin.
Since
P(outcome ∈ A or outcome ∈ B) = P(outcome ∈ A ∪ B),
P(outcome ∈ A and outcome ∈ B) = P(outcome ∈ A ∩ B),
and
P(outcome ∉ A) = P(outcome ∈ A')
where A' denotes the complementary set to A, the above rules for
probabilities correspond to the following rules for μ, called
axioms :

μ(Ω) = 1, where Ω is the set of all possible outcomes,
and, if μ(A) and μ(B) are defined, then so are μ(A'),
μ(A∪B), and μ(A∩B).
 μ(A) ≥ 0.
 μ(A∪B) = μ(A)+ μ(B) if A and B are disjoint.
Probability theory
Luckily, it turns out,
Kolmogorov (1933)
calls this the Fundamental Theorem,
that practically all μ's encountered in
practice satisfy the following:
Such a μ is called a probability measure defined on an
algebra of subsets of a given set.
The Extension Theorem says that μ's
domain of definition can be
extended so that one remains within it even after forming infinite
unions and intersections. The extended μ is called a probability
measure defined on a σalgebra . This implies:
Link to Kolmogorov's
"Grundbegriffe der Wahrscheinlichkeitsrechnung" (1933),
translated to English, at www.mathematik.com
Kolmogorov's Fundamental Theorem of Probability
Calculus
Bengt Ringnér
Centre for Mathematical Sciences, Lund University, Lund, Sweden
http://www.maths.lth.se/matstat/staff/bengtr