092909A Lecture on Probability

092909A Lecture on Probability - A Lecture on Probability...

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Lecture on Probability A Lecture on Probability Rob Leachman 10 E10 September 29, 2009 Sep 29, 2009 Basic Probability 1
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ncertainty Uncertainty lmost all decisions involve some level of Almost all decisions involve some level of uncertainty ategories of uncertainty Categories of uncertainty – Unknown but knowable for some cost and effort (e.g., overconfidence test) – Some residual level of uncertainty, but we can describe range and likelihood – Completely unknown • Probability is suited for category 2 Sep 29, 2009 Basic Probability 2
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robability theory Probability theory robability theory grew out of gambling Probability theory grew out of gambling, which dates at least as far back as ancient gypt Egypt • First formal probability theory was ublished by Pascal circa 1700 published by Pascal circa 1700 • A useful pictorial aid for thinking about probability is the Venn Diagram • Let S denote all possible outcomes Sep 29, 2009 Basic Probability 3
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robability Rules Probability Rules • Let A and B denote particular subsets of the set f all possible outcomes The probability of of all possible outcomes S . The probability of A, which we denote as P ( A ), is the fraction of total utcomes in at belong to outcomes in S that belong to A. • By definition, P ( S ) = 1. Moreover, 1 ) ( 0 A P • If A is a subset of B, then • The set of outcomes not in A is called the ) ( ) ( B P A P complement of A , denoted by A c . Note that P ( A c ) = 1 – P ( A ). (This is called the subtraction rule.) Sep 29, 2009 Basic Probability 4
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enn Diagram Venn Diagram S A B As drawn, A and B have some outcomes common in common. P{A or B} = P{A and not B} = P{A} + P{B} – P{A and B} P{A} – P{A and B} Sep 29, 2009 Basic Probability 5
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robability Rules Probability Rules • The addition rule is P ( A or B ) = P ( A ) + P ( B ) – P ( A and B ). In particular, if A and B re utually exclusive .e., no events in are mutually exclusive (i.e., no events in common), then P ( A or B ) = P ( A ) + P ( B ). • Example: Roll a die. A = {2, 4, 6}, i.e., we get an even number. B = {1, 2, 3}. Then P(A or B ) = 3/6 + 3/6 – 1/6 = 5/6. Sep 29, 2009 Basic Probability 6
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nother Example Another Example • Suppose 50% of beer drinkers like Anchor team 40% like Becks 30% like both Steam, 40% like Becks, 30% like both. • Let P ( A ) = probability a random beer rinker likes Anchor Steam = drinker likes Anchor Steam, P ( B ) = probability a random beer drinker likes ecks We are given =05 = Becks. We are given P ( A ) 0.5, P ( B ) 0.4, P ( A and B ) = 0.3. hat is the probability a random drinker What is the probability a random drinker likes either one? P(A r B = 0.5 + 0.4 .3 = 0.6 Sep 29, 2009 Basic Probability 7 ( o )0 50 03 06
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xample (cont ) Example (cont.) • What is the probability a random drinker likes Anchor Steam but not Becks?
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092909A Lecture on Probability - A Lecture on Probability...

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