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Probability    

Random experiment: process leading to an uncertain outcome Basic Outcome: Possible outcome of a random income Sample Space: Collection of all the possible outcomes of a random experiment Event: Any subset of basic outcomes from the sample space

Events Intersection of Events Being A and B two event in a sample space, the intersection (A∩B) is the set of all outcomes in S that belong to both

Mutually exclusive events If the events have no basic outcomes in common: set A∩B is empty

Union of Events Being A and B two event in a sample space, the union (AUB) is the set of all outcomes in S belonging to either A or B

Collective exhaustive events If the union of all the events is the sample space: AUBUCUD…UN=S

Complement The complement of an event (A) is the set of all basic outcomes in the simple space that do not belong to it ()

Probability Probability of an uncertain event to occur will always be between 0 (impossible) and 1 (certain)

Assessing probability Approaches to assessing the probability to an uncertain event: 

Classical probability

P (A) = Number of events that satisfy A / total number of events It assumes all basic outcomes in the sample spaces are equally likely 

Relative frequency or empirical probability

P (A) = Number of times that satisfy A in N trials / number of trials What we observe after repeating an experiment N times: As more N, more near to classical probability



Subjective probability

Individual’s opinion about an event

Probability postulates If A is any event in the sample space: 0≤ P(A) ≤1 If A is any event in S and  are the basic outcomes: P(A) = ∑    Probability of S will always be 1: P(S) = 1

Probability Rules Complement Rule:

P() = 1- P(A)

Addition Rule:

P(AUB) = P(A) + P(B) – P(A∩B)

Conditional probability Probability of an event given another event that has occurred: P (A|B) = P(A∩B)/P(B) P (B|A) = P(A∩B)/P(A)

Multiplication Rule From the definition of the conditional probability we can get the probability of the intersection of two events: P(A∩B) = P(A|B) · P(B)

or

P(A∩B) = P(A|B) · P(A)

Statistical independence Two events are statistically independent if: P (A∩B) = P(A) · P(B) The probability of one event is not affected by the other event: P(A|B) = P(A)

If P(B)˃o

P(A|B) = P(B)

If P(A)˃0...