Slides Chapter 1. Measure Theory and Probability

1.5. OTHER DEFINITIONS OF PROBABILITY
1.5 Other definitions of probability
WhenΩisfinite,sayΩ = {a 1 ,...,a k },manytimestheelementary
events are equiprobable, that is, P({a 1 }) = ··· = P({a k }) = 1/k.
Then, for A ⊂ Ω, say A = {a i1 ,...,a im }, then
m∑
P(A) = P({a ij }) = m k .
j=1
This is the definition of probability given by Laplace, which is
useful only for experiments with a finite number of possible results
and whose results are, a priori, equally frequent.
Definition 1.22 (Laplace rule of probability)
The Laplace probability of an event A ⊆ Ω is the proportion of
results favorable to A; that is, if k is the number of possible results
or cardinal of Ω and k(A) is the number of results contained in A
or cardinal of A, then
P(A) = k(A)
k .
In order to apply the Laplace rule, we need to learn to count.
The counting techniques are comprised in the area of Combinatorial
Analysis.
The following examples show intuitively the frequentist definition
of probability.
Example 1.15 (Frequentist definition of probability)
The following tables report the relative frequencies of the results
of the experiments described in Example 1.8, when each of these
experiments are repeated n times.
(a) Tossing a coin n times. The table shows that both frequencies
of “head” and “tail” converge to 0.5.
ISABEL MOLINA 19