02.1 Combinatorial Analysis

# 02.1 Combinatorial Analysis - Axioms of Probability Axiom 1...

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1/5/10 Axioms of Probability Axiom 1 0 ° Axiom 2 P(S) = 1 Axiom 3 = = = = 1 1 ) ( when i i i i j i E P E P j i E E j Axiom 3 in words. Given that 2 events A and B are M.E., the probability of at least 1 of these events is the sum of their probabilities. Pr{A“B} = Pr{A ‘or’ B} = Pr{A} +

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1/5/10 Axiom 3 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 To answer part b) first, we will assume the events: To answer part a) Convert "odds"
1/5/10 Axiom 3, again Let A be an event, and let a1 , a2, ... , an be the outcomes (elements) comprising 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. Example Flip 4 coins, find the probability that

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1/5/10 Simple Propositions Proposition 1 Proposition 2 ) ( ) ( then , If, F P E P F E Proposition 3 ) ( ) ( ) ( ) ( F E P F P E P F E P - + =
1/5/10 Proposition 1 example Based on the curve used in a class, the probabilities of grades are: Pr{A or A-} = 0.25 Pr{B+, B, or B-} = 0.3 Answer: ) ( 1 ) ( E P E P c - =

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1/5/10 Proposition 3 Example. If we roll a die, find the probability of an odd number or a number greater than 2. 1) Let event A = "odd number" = {1,3,5} 2) Let event B = "number greater than 2" = { 3, 4, 5, 6} ) ( ) ( ) ( ) ( F E P F P E P F E P - + =
1/5/10 Combinatorial Analysis Communication system of n identical antennas, lined up linearly

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