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Lecture 9

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McMaster University

Economics

ECON 2B03

Jeff Racine

Fall

Description

Lecture 9
P(A)=P(A∩B)+P(A∩B) ́
P(A∪B)=P(A)+P(B)−P(A∩B)
P(A∣B)= P(A∩B) ∨P(B)×P(A ∣B)=P(A∩B)
P(B)
P(B∩A)
P(B∣A)= ∨P(A)×P(B ∣A)=P(B∩ A)
P(A)
¿P(B)×P(A ∣B)
P(A)
Example (unconditional, conditional, joint probability)
The probability of drawing a heart (event B) is an unconditional probability
p(heart) = p(B) = 13/52
The probability of drawing a face card (event A) given tht the card drawn was a heart
(event B) is a conditional probability
p(face card| heart) = p(A|B) = (3/52)/(13/52) = 3/13
Now, suppose you had the et of joint probabilities, say, p(face card and heart), p(face
card and club), p(face card and spade), and p(face card and diamond), and you needed
to compute the unconditional probability of, say, obtaining a face card
You could apply the unconditional from joint probability rule, i.e.,
P(face card) = all suitsecard∩suits) = p(face card and heart) +p(face card and
club)+p(face card and spade) + p(face card and diamond)
3/52 + 3/52 + 3/52 + 3/52 = 12/52
Joint Probability Tables
A joint probability table shows frequencies or relative frequencies for joint events
Laws of Probability: Multiplication
The General Multiplication Law (probability of A and B) The joint probability of two events happening at the same time equals the unconditional
probability of one event times the conditional probability of the other event, given that the
first event has already occurred
p(A∩B)=p(A)× p(B ∣A)=p(B)×p(A ∣B)
The events A and B are dependent when the probability of occurrence of A is
affected by the occurrence of B, hence p(A)≠ p(A∨B) . Independent events are
p(A)=p(A∨B)
those for which
The Special Multiplication Law (multiplication law for independent events)
p(A∩B)=p(A)× p(B)
Bayes’ Theorem (Reverend Thomas Bayes (1702 – 1761))
How do we estimate or revise probabilities when new information is obtained? By using
Bayes’ Theorem!
Consider a random experiment having several possible mutually exclusive outcome E , 1
…,E n Suppose that the probabilities of these outcomes, p(E) iave been obtained
We call these Prior Probabilities since they are obtained before new information is
taken into account
A revised probability is called a Posterior Probability
Posterior probabilities are conditional probabilities, the conditioning event being new
information
By using Bayes’ Theorem, a prior probability is modified
The new probability is a posterior (conditional) probability
Let A be the new information. By the definition of conditional probability, the posterior
prob

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