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Unformatted text preview: 11COMP170Discrete Mathematical Toolsfor Computer ScienceDiscrete Math for Computer ScienceK. Bogart, C. Stein and R.L. DrysdaleSection 5.3, pp. 236247IndependenceVersion 2.0: Last updated, May 13, 2007Slidesc2005 by M. J. Golin and G. Trippen21Conditional Probability and Independence•Conditional Probability•Independent Trials Processes•Independence31Conditional ProbabilitySuppose we’ve thrown two fair dice. Theprobability of seeing “doubletwos” is136.32Conditional ProbabilitySuppose we’ve thrown two fair dice. Theprobability of seeing “doubletwos” is136.Now suppose that we don’t seethe dice but know that the event“the dice sum up to4”has occured. What is the probabilitythat “doubletwos” occurredgiven that“the dice sum up to4”?{,,33Conditional ProbabilitySuppose we’ve thrown two fair dice. Theprobability of seeing “doubletwos” is136.Now suppose that we don’t seethe dice but know that the event“the dice sum up to4”has occured. What is the probabilitythat “doubletwos” occurredgiven that“the dice sum up to4”?{,,Answer “should be”13, shouldn’t it?34Conditional ProbabilitySuppose we’ve thrown two fair dice. Theprobability of seeing “doubletwos” is136.Now suppose that we don’t seethe dice but know that the event“the dice sum up to4”has occured. What is the probabilitythat “doubletwos” occurredgiven that“the dice sum up to4”?{,,Answer “should be”13, shouldn’t it?This lecture formalizes this intuition.41A more complicated example422 dice:A more complicated example432 dice:Event”at least one circle on top”is:{},,A more complicated example,,442 dice:Event”at least one circle on top”is:{},,Applyingprinciple of inclusion and exclusion: probability ofseeing a circle on at least one top when we roll the dice is13+1319=59A more complicated example,,51Suppose you are told that the two dice have beenrolled andboth top shapes are the same?52Suppose you are told that the two dice have beenrolled andboth top shapes are the same?What is the probability that at least one top shape(and now therefore both top shapes)is a circle?53Suppose you are told that the two dice have beenrolled andboth top shapes are the same?What is the probability that at least one top shape(and now therefore both top shapes)is a circle?Originally, (i) chance of getting (two) circles was4timeschance of getting (two) triangles and (ii) chance of getting(two) squares was9times chance of getting (two) trianglesso,given that both top shapes are the sameintuitively, we“should ”have54Suppose you are told that the two dice have beenrolled andboth top shapes are the same?What is the probability that at least one top shape(and now therefore both top shapes)is a circle?...
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 Spring '10
 M.J.Golin
 Computer Science

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