02-addition-combin-1

02-addition-combin-1 - IE 111 Fall Semester 2011 Note Set...

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IE 111 Fall Semester 2011 Note Set #2 3.5 Elementary Theorems of Probability Recall the 3 basic Axioms of probability are: 1. For all events A in any sample space S (i.e. for all subsets A of S) 0 Pr{A} 1.0 2. Pr{S} = 1.0 3. If events A and B are mutually exclusive, then: Pr{A or B} = Pr{A "union" B} = Pr{A} + Pr{B} From these axioms, some important elementary theorems can be easily obtained. Theorem 1. If A, B, and C are mutually exclusive events then Pr{A or B or C} = Pr{A} + Pr{B} + Pr{c} This is easily proven since (A or B or C) = ((A or B) or C) This also then extends to n mutually exclusive events: Pr{ A or B or C or D or . ...} = Pr{A} + Pr{B} + Pr{C} + Pr{D} + . .... (Remember that "or" is the same as "union") Example There are 3 football teams from Pennsylvania in the Patriot League. Odds makers give the following odds on each of these teams winning the league championship: Lehigh: 3 to 1 Bucknell: 5 to 1 Lafayette: 100 to 1 a) Calculate the probability that a Pennsylvania team will win the Patriot League title. b) What assumption is necessary in this calculation? To answer part b) first, we will assume the events (Lehigh wins), (Bucknell wins), (Lafayette wins) are Mutually Exclusive. In this case it means we assume no ties for league champion. To answer part a) we first convert "odds" to probabilities:
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Pr{Lehigh} = 1/(3+1) = 0.25 Pr{Bucknell} = 1/(5+1) = 0.1666 Pr{Lafayette} = 1/(100+1) = 0.0099 By Theorem 1. Pr{Pennsylvania team wins} = 0.25 + 0.1666 + 0.0099 = 0.4266 Theorem 2. Pr{A} = 1 - Pr{A } (where A is the "complement of A" or "NOT A") This is clearly true due to the following three known facts: 1. S = {A} union {A } 2. Pr{S} = 1.0 3. A and A are clearly M.E. Example Based on the curve used in this class, your probabilities of grades are: Pr{A or A-} = 0.25 Pr{B+, B, or B-} = 0.3 Pr{C+,C,or C-} = 0.35 Find Pr{D+,D,D-,or F} Answer: The events {A or A-}, {B+, B, or B-}, and {C+,C,or C-} are clearly mutually exclusive (You must get only one grade). Thus the probability of a C- or better is 0.25+0.3+0.35 = 0.9 (from theorem 1.) The event {D+,D,D-,or F} is the same as {NOT C- or better} We have Pr{S} = Pr{C-or better} + Pr{NOT C- or better} 1.0 = 0.9 + Pr{NOT C- or better} Thus Pr{NOT C- or better} = 1.0 - 0.9 = 0.1 (from theorem 2.)
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Theorem 3. Let A be an event, and let a 1 , a 2 , . .. , a n be the outcomes (elements) comprising event A Then Pr{A} = Pr{ a 1 } + Pr{ a 2 } + . .. + Pr{ a n } This is clearly true since each outcome is an element of S and thus a subset of S and an event. Also, outcomes are , by definition , M.E. since they are elements of S. Thus by theorem 1, theorem 2 must be true. Example Flip 4 coins, find the probability that exactly 2 are heads. There are 2
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02-addition-combin-1 - IE 111 Fall Semester 2011 Note Set...

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