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Lecture 9

Lecture 9.docx

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McMaster University
Jeff Racine

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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