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2. Probability
Probabilité et Statistique I — Section 2.1
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Petit Vocabulaire
Mathematics
English
Français
set
un ensemble
A∪B
union
l’union
A∩B
Ac
intersection
l’intersection
complement of A (in Ω)
le complémentaire de A (en Ω)
A \ B
difference
la différence
A ∆ B
symmetric difference
la différence symmétrique
A×B
Cartesian product
le produit cartesien
cardinality
{Aj }n
pairwise disjoint
j=1
partition
le cardinal
{Aj }n
disjoint deux à deux
j=1
une partition
permutation
une permutation
combination
une combinaison
binomial coefficient
r)
un coéfficient binomial (Cn
multinomial coefficient
un coéfficient multinomial
indistinguishable
indifférentiable
colour-blind
daltonien (ienne)
Ω, A, B . . .
|A|
n
r
n
n1 ,...,nr
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Petit Vocabulaire Probabiliste
Mathematics
English
Français
one fair die (several fair dice)
un dé juste (plusieurs dés justes)
random experiment
une expérience aléatoire
Ω
sample space
l’ensemble fondamental
ω
outcome, elementary event
une épreuve, un événement élémentaire
event
un événement
event space
l’espace des événements
sigma-algebra
une tribu
probability distribution/probability function
une loi de probabilité
probability space
un espace de probabilité
inclusion-exclusion formulae
formule d’inclusion-exclusion
probability of A given B
la probabilité de A sachant B
independence
indépendance
(mutually) independent events
les événements (mutuellement) indépendants
pairwise independent events
les événements indépendants deux à deux
conditionally independent events
les événements conditionellement indépendants
A, B, . . .
F
P
(Ω, F , P)
P(A | B)
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2.1 Probability Spaces
Contents
Elementary examples of probability, motivation
Ideas of random experiment, probability space, sample space, event
space, probability measure, inclusion-exclusion formulae
References: Ross (Chapter 2); Ben Arous notes (II.1–II.3).
Exercises: 19–36 of Recueil d’exercices.
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Motivation: Game of Dice
Example (Two dice): Two fair dice are rolled, one red and one
green.
(a) What is the set of all possible outcomes?
(b) Which outcomes give a total of 6?
(c) Which outcomes give a total of 12?
(d) Which outcomes give an even total?
(e) What are the probabilities of the events in (b), (c), (d)?
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Computation of Probabilities
Could try and compute probabilities for events such as (b), (c), (d)
by rolling the dice many times, and setting
probability of event =
# times event occurs
.
# times experiment performed
But this is a physical answer, rather than a mathematical one, not
always possible, only ever available after a lot of work (how many
times must the dice be rolled?), and liable to give different answers
each time — unsatisfactory!
Probabilities in simple examples can often be computed using
symmetry. This fails in more complicated cases — must build
mathematical model, based on the ideas of a random experiment
and probability space.
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Random Experiment
Definition: A random experiment (une expérience aléatoire)
is an ‘experiment’ whose outcome is (or can be treated as) random.
Example 2.1: I toss a coin.
Example 2.2: The number of punctures I get going home.
Example 2.3: I roll two fair dice, one red and one green.
Example 2.4: The number of emails I receive today.
Example 2.5: The waiting time to the end of this lecture.
Example 2.6: The change in the Swiss stock market (SMI) in 2003.
Example 2.7: The weather here at noon tomorrow.
A random experiment is modelled by a probability space (un
espace de probabilité).
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Probability Space (Ω, F, P)
Definition: A probability space (Ω, F, P) is a mathematical
object associated to a random experiment, consisting of three things:
• a set Ω, the sample space (ensemble fondamental ), which
contains all the possible outcomes (épreuves, événements
élémentaires) of the experiment;
• a collection F of subsets of Ω. These subsets are called events
(événements), and F is called the event space (l’espace des
événements);
• a function P : F 7→ [0, 1] called a probability function (loi de
probabilité), which associates a probability P(A) to each A ∈ F.
We look at each of these in turn.
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Sample Space Ω
The sample space Ω is a set whose elements represent all the possible
outcomes for a random experiment. Each element ω ∈ Ω is associated
to a distinct outcome.
Ω is analogous to the universal set.
Ω is non-empty. (If Ω = ∅ then nothing interesting can happen.)
Ω can be finite, countable, or uncountable.
Exercise: Write down the sample spaces for Examples 2.1–2.7.
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Event Space F
F is a set of subsets of Ω which represent the events of interest.
Example 2.3 (ctd): Consider the events A ‘the red die shows 4’, B
‘the total is even’, C ‘the green die shows 2’, and A ∩ B ‘the red die
shows 4 and the total is even’.
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Definition: An event space F is a set of subsets of Ω such that:
1. F is non-empty;
2. if A ∈ F then Ac ∈ F;
3. if
{Ai }∞
i=1
are all members of F, then
S∞
i=1
Ai ∈ F.
F is also called a sigma-algebra (une tribu).
Exercise: Write down event spaces for Examples 2.1–2.4.
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Event Space F, II
Let A, B, C, {Ai }∞
i=1 be members of F. The axioms for F imply that
Sn
• i=1 Ai ∈ F,
• Ω ∈ F,
∅ ∈ F,
• A ∩ B ∈ F, A \ B ∈ F,
Tn
• i=1 Ai ∈ F.
A ∆ B ∈ F,
If Ω is countable, F is often taken to be the set of all subsets of Ω.
This is the largest (and richest) possible event space for Ω.
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Event Space F, III
Different event spaces can be defined for the same sample space.
Example 2.3 (ctd): I roll two fair dice, one red and one green.
(a) What is my event space F1 ?
(b) I tell my friend only the total. What is his event space F2 ?
(c) My friend looks at the dice himself, but he is colour-blind. What
is now his event space F3 ?
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Usually the event space is clear from the context, but it is important
to write both Ω and F out explicitly, in order to avoid confusion. It
can also be helpful when so-called ‘paradoxes’ arise (usually because
of an unclear or faulty mathematical formulation of the problem).
It is essential to give Ω and F in exercises, tests, and exams.
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Three Dice Problem — Galileo (1564–1642)
Three fair dice are thrown. Let Ti be the event ‘the total is i’, for
i = 3, . . . , 18. Which is the more probable, T9 or T10 ?
T9 can occur if the dice show
(6, 2, 1), (5, 3, 1), (5, 2, 2), (4, 4, 1), (4, 3, 2), (3, 3, 3).
T10 can occur if the dice show
(6, 3, 1), (6, 2, 2), (5, 4, 1), (5, 3, 2), (4, 4, 2), (4, 3, 3).
So they are equally probable.
True or false?
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Example 2.8 (Family planning): A woman planning her family
considers the following schemes, on the assumption that boys and
girls are equally likely on each delivery:
(a) have three children;
(b) bear children until the first girl is born, or until three are born,
whichever is sooner, and then stop;
(c) bear children until there is one of each sex or until there are
three, whichever is sooner, and then stop.
Let Bi denote the event that i boys are born, and let C denote the
event that there are more girls than boys. Find P(B1 ) and P(C)
under the schemes above.
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Probability Function P
Definition: A probability function P associates a probability to
each element of an event space F, with the properties:
1. if A ∈ F, then 0 ≤ P(A) ≤ 1;
2. P(Ω) = 1;
3. if {Ai }∞
i=1 are pairwise disjoint (that is, Ai ∩ Aj = ∅, i 6= j), then
!
∞
∞
[
X
P
Ai =
P(Ai ).
i=1
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Properties of P
Theorem 2.1: Let A, B, {Ai }∞
i=1 be events of a probability space
(Ω, F, P). Then
(a) P(∅) = 0,
(b) if A ∩ B = ∅, then P(A ∪ B) = P(A) + P(B),
(c) P(Ac ) = 1 − P(A),
P(A ∪ B) = P(A) + P(B) − P(A ∩ B),
(d) if A ⊂ B, then P(A) ≤ P(B), and P(A \ B) = P(A) − P(B)
S∞
P∞
(e) P ( i=1 Ai ) ≤ i=1 P(Ai ) (Boole’s inequality),
S∞
(f) if A1 ⊂ A2 ⊂ · · ·, then limn→∞ P(An ) = P ( i=1 Ai ),
T∞
(g) if A1 ⊃ A2 ⊃ · · ·, then limn→∞ P(An ) = P ( i=1 Ai ).
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Continuity of P
Recall that a function f is continuous at x if for any sequence {xn }
such that
lim xn = x, we have lim f (xn ) = g(x).
n→∞
n→∞
Parts of (f) and (g) of Theorem 2.1 can be extended to show that for
any sequences of sets for which
lim An = A, we have lim P(An ) = P(A).
n→∞
n→∞
Hence P is called a continuous set function.
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Inclusion-Exclusion Formulae
If A1 , . . . , An are events of a probability space (Ω, F, P ), then
P(A1 ∪ A2 ) = P(A1 ) + P(A2 ) − P(A1 ∩ A2 )
P(A1 ∪ A2 ∪ A3 ) = P(A1 ) + P(A2 ) + P(A3 )
−P(A1 ∩ A2 ) − P(A1 ∩ A3 ) − P(A2 ∩ A3 )
+P(A1 ∩ A2 ∩ A3 )
..
.
P
n
[
i=1
Ai
!
=
n
X
r=1
(−1)
r+1
X
P(A1 ∩ · · · ∩ Air ).
1≤i1 <···<ir ≤n
The number of terms in the general formula is
n
n
n
n
n
+
+
+ ···+
+
= 2n − 1.
1
2
3
n−1
n
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Example 2.9: What is the probability of at least one 6 when I roll
three fair dice?
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Example 2.10: An urn contains 1000 lottery tickets numbered
from 1 to 1000. One is selected at random. A fairground performer
offers to pay $3 to anyone who has aleady paid him $2, if the number
on the ticket is divisible by 2, 3 or 5. Would you pay him your $2
before the draw? (You will lose your money if the ticket number is
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not divisible by 2, 3, or 5.)
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