L16_Independence

L16_Independence - 1-1COMP170Discrete Mathematical Toolsfor...

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Unformatted text preview: 1-1COMP170Discrete Mathematical Toolsfor Computer ScienceDiscrete Math for Computer ScienceK. Bogart, C. Stein and R.L. DrysdaleSection 5.3, pp. 236-247IndependenceVersion 2.0: Last updated, May 13, 2007Slidesc2005 by M. J. Golin and G. Trippen2-1Conditional Probability and Independence•Conditional Probability•Independent Trials Processes•Independence3-1Conditional ProbabilitySuppose we’ve thrown two fair dice. Theprobability of seeing “double-twos” is136.3-2Conditional ProbabilitySuppose we’ve thrown two fair dice. Theprobability of seeing “double-twos” 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 “double-twos” occurredgiven that“the dice sum up to4”?{,,3-3Conditional ProbabilitySuppose we’ve thrown two fair dice. Theprobability of seeing “double-twos” 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 “double-twos” occurredgiven that“the dice sum up to4”?{,,Answer “should be”13, shouldn’t it?3-4Conditional ProbabilitySuppose we’ve thrown two fair dice. Theprobability of seeing “double-twos” 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 “double-twos” occurredgiven that“the dice sum up to4”?{,,Answer “should be”13, shouldn’t it?This lecture formalizes this intuition.4-1A more complicated example4-22 dice:A more complicated example4-32 dice:Event”at least one circle on top”is:{},,A more complicated example,,4-42 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+13-19=59A more complicated example,,5-1Suppose you are told that the two dice have beenrolled andboth top shapes are the same?5-2Suppose 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?5-3Suppose 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 ”have5-4Suppose 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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L16_Independence - 1-1COMP170Discrete Mathematical Toolsfor...

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