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Think of the simpler case in which a and b are

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Think of the simpler case in which A and B are mutually exclusive. That is, they cannot bothoccur. Then P(A∩B), the probability that they occur together, is 0. So the last part of the equationcan be dropped for this special case. We end up with:P(AB) = P(A) + P(B)Downloaded by Tori Lee ([email protected])lOMoARcPSD|3298179
The outcome (AB) occurs just in case either one of A or B occurs. So P(AB) is just theprobability of A plus the probability of B.Adding the probabilities is not only correct, but can be made intuitive. Which is likelier: that Aoccurs, or that any one of A, B or C occurs?lOMoARcPSD|3298179
P(A∩B) = P(A) x P(B)2.DependentAandB; the probability thatAoccurs is affected byB’soccurring.Claims:Premise is descriptive claim and conclusion is a normative claim.P(A∩B) = P(A|B) x P(B)P(A|B)’ is a conditional probability:the probability ofAgivenB.Plausibly, whether Venus is aligned with Neptune is independent of whether Ted eventually suffers fromlung cancer.So,P(A∩B) = P(A) x P(B)We just multiply the independent probabilities of these two events.By contrast, suppose we want to know the probability of the scenario in which Ted smokes cigarettes andTed eventually suffers from lung cancer.Probability that Ted suffers from lung cancer ≈ .0007Probability that Ted smokes ≈ .22If we treated these as independent events, we would just multiply the probabilities:P(L∩S) = P(L) x P(S) = .0007 x .22 = .00015…or about 15 in 100,000.Downloaded by Tori Lee ([email protected])lOMoARcPSD|3298179
But this overlooks something important: the probabilities of having lung cancer and of being asmoker are dependent upon each other. If one smokes, one is much more likely to get lungcancer; and if one gets lung cancer, one is much more likely to have smoked.The basic idea goes back to the truth-conditions of conditional statements.For example:Suppose we want to know whether S is both a fox and a mammal.Does it make a difference to know that if S is a fox then S is a mammal?

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Term
Winter
Professor
Cratsley
Tags
Logic, Tori Lee

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