Chapter 1
PROBABILITES Reminders
1.1 Probabilities and events
Objective
At the end of these two sessions of 4 hours each, the Student who has applied himself
will be able to:
1. Define an event, a probability and enumerate the basic axioms of a tribe of events;
2. To apply the axiomatic of a tribe of events and the definition of a probability to solve
the application exercises and certain exercises and problems of deepening relating
thereto;
3. Define the concepts of conditional probabilities and recognize the type of problems
that these concepts model;
4. To distinguish the Formula of Total Probabilities from that of Thomas Bayes;
(a) To recognize the problems modeled by conditional probabilities and to represent
them by a weighted tree;
(b) to recognize and solve the application exercises and certain problems of deepening
relating to the formulas of Total Probabilities and that of Thomas Bayes.
1.1.1 Basics
Definition 1:
A random experiment is any experiment whose outcome cannot be accurately predicted but
for which we know all the Ω of all possible outcomes.
The examples are legion.
1. The jet of a cubic die is a random experiment whose fundamental set is Ω = {1, 2, 3, 4,
5, 6}.
2. The throwing of a coin is a random experiment whose fundamental set is Ω = {pile,
face}
3. If the experiment consists of two successive jets of a coin we have the fundamental set:
is
Ω = {(p, p), (p, f), (f, p), (f, f)}
, 4. Following the same logic, two successive throws of a balanced die correspond to the
fundamental whole:
Ω = {(,). (,) , ··· , (,)}
For this fundamental set, the issue (i, j) corresponds to obtaining the number i for the
first draft and the number j for the second.
5. The experiment of measuring the operating time of a computer before the first failure
occurs corresponds to the fundamental set Ω = [0, +∞ [
Some facts related to any random experience may or may not occur. They are called
events.
It is known that for any random experiment which is associated with the fundamental set
Ω, an event is always a subset of the fundamental set Ω.
Thus, for example,
1. The event {f} corresponds to the appearance from the front after the throwing of a coin,
2. The event E = {2, 4, 6} corresponds to the appearance of an even number after the
throwing of a die,
3. The event E = {(p, p), (p, f)} corresponds to the appearance of a pile for the first jet
during the experiment consisting of two successive jets of a cubic die.
4. The event E = {(,), (,), (,), (,), (,)} corresponds, for example, to obtaining a sum of 7
when adding the respective numbers obtained for the two throws of a die
It is known that for a random experiment corresponding to the fundamental set Ω, if
E1 and E2 are events then, the conjunction E1 ∩ E2 and the disjunction E1 ∪ E2 of these
¯
events are also events, in addition, any event A corresponds to the event opposite A ,
with,
Ω and ∅ are special events:
∅ is the impossible event because it has no chance of
happening; Ω is the certain event because it always happens.
If E1 ∩ E2 = ∅ then events E1 and E2 are said to be incompatible.
Definition 2:
For a random experiment corresponding to the fundamental set Ω,
An event tribe is a subset of the set P (Ω) of parts of Ω such that:
;
We
have:
PROBABILITES Reminders
1.1 Probabilities and events
Objective
At the end of these two sessions of 4 hours each, the Student who has applied himself
will be able to:
1. Define an event, a probability and enumerate the basic axioms of a tribe of events;
2. To apply the axiomatic of a tribe of events and the definition of a probability to solve
the application exercises and certain exercises and problems of deepening relating
thereto;
3. Define the concepts of conditional probabilities and recognize the type of problems
that these concepts model;
4. To distinguish the Formula of Total Probabilities from that of Thomas Bayes;
(a) To recognize the problems modeled by conditional probabilities and to represent
them by a weighted tree;
(b) to recognize and solve the application exercises and certain problems of deepening
relating to the formulas of Total Probabilities and that of Thomas Bayes.
1.1.1 Basics
Definition 1:
A random experiment is any experiment whose outcome cannot be accurately predicted but
for which we know all the Ω of all possible outcomes.
The examples are legion.
1. The jet of a cubic die is a random experiment whose fundamental set is Ω = {1, 2, 3, 4,
5, 6}.
2. The throwing of a coin is a random experiment whose fundamental set is Ω = {pile,
face}
3. If the experiment consists of two successive jets of a coin we have the fundamental set:
is
Ω = {(p, p), (p, f), (f, p), (f, f)}
, 4. Following the same logic, two successive throws of a balanced die correspond to the
fundamental whole:
Ω = {(,). (,) , ··· , (,)}
For this fundamental set, the issue (i, j) corresponds to obtaining the number i for the
first draft and the number j for the second.
5. The experiment of measuring the operating time of a computer before the first failure
occurs corresponds to the fundamental set Ω = [0, +∞ [
Some facts related to any random experience may or may not occur. They are called
events.
It is known that for any random experiment which is associated with the fundamental set
Ω, an event is always a subset of the fundamental set Ω.
Thus, for example,
1. The event {f} corresponds to the appearance from the front after the throwing of a coin,
2. The event E = {2, 4, 6} corresponds to the appearance of an even number after the
throwing of a die,
3. The event E = {(p, p), (p, f)} corresponds to the appearance of a pile for the first jet
during the experiment consisting of two successive jets of a cubic die.
4. The event E = {(,), (,), (,), (,), (,)} corresponds, for example, to obtaining a sum of 7
when adding the respective numbers obtained for the two throws of a die
It is known that for a random experiment corresponding to the fundamental set Ω, if
E1 and E2 are events then, the conjunction E1 ∩ E2 and the disjunction E1 ∪ E2 of these
¯
events are also events, in addition, any event A corresponds to the event opposite A ,
with,
Ω and ∅ are special events:
∅ is the impossible event because it has no chance of
happening; Ω is the certain event because it always happens.
If E1 ∩ E2 = ∅ then events E1 and E2 are said to be incompatible.
Definition 2:
For a random experiment corresponding to the fundamental set Ω,
An event tribe is a subset of the set P (Ω) of parts of Ω such that:
;
We
have:
