web14 - Probability 1 Todays plan Probability Notations...

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1 Probability
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2 Today’s plan Probability Notations Laws of probability
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3 Set Notation Sample space All possible outcomes of an experiment Example: You are playing a roulette with 100 numbers Set S consists of each of the 100 possible outcomes labeled 1,2,… 100: S={1,2,…,100} This is the sample space Examples of sample space: Toss a coin once S={H,T} Roll a dice once S={1,2,3,4,5,6} Toss a coin twice S={HH,HT,TH,TT} Gender of a baby S={boy,girl} All letters of the alphabet S={a,b,c,…,z}
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4 Set Notation Event A subset of the sample space S={1,2,3,4,5,6} Subset A is the odd numbers A={1,3,5} A is a subset of S: A S 3 is an element of A: 3 A
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5 Example of an event Event A = getting exactly one head when tossing a coin twice S={HH,HT,TH,TT} A={HT,TH}
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6 Venn diagram A tool for describing relations between sets A B S
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7 Venn diagram Sample space: S={1,2,3,4,5,6} A={1,3,5} B={2,4} A B S
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8 Venn diagram Sample space: S={1,2,3,4,5,6} A={1,3,5} B={1,2,4} A B S
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9 = Complement of A . Includes all outcomes in S that are not in A Also denoted by A c 2200 φ = An empty set Complement of a set A ? = S A A S φ
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10 Example: S={1,2,3,4,5,6} A={1,3,5} {2,4,6} = A A A S
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11 Union of sets is the union of sets A and B. The set of outcomes that are in A or B. The event that either A, B, or both occur. B A Dice example: S={1,2,3,4,5,6} A={1,3,5} B={1,2} } 5 , 3 , 2 , 1 { = B A A B S
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12 Example S={1,2,3,4,5,6,7,8,9} A={2,4,6,8} B={1,3,5,9} C={1,2,3} AUC= AUB= AUB c = AUC c = {1,2,3,4,6,8} {1,2,3,4,5,6,8,9} {2,4,6,7,8} {2,4,5,6,7,8,9}
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13 Intersection of sets is the intersection of sets A and B. The set of outcomes that are in A and B. The event that both A and B occur. B A } 1 { = B A A B B A S Dice example: S={1,2,3,4,5,6} A={1,3,5} B={1,2}
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14 Example S={1,2,3,4,5,6,7,8,9} A={2,4,6,8} B={1,3,5,9} C={1,2,3} A∩B= A∩C= A∩B c = A c ∩C= {2} {2,4,6,8} {1,3} φ
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15 Disjoint sets A and B are disjoint sets if φ = B A A B S
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16 Example: S={1,2,3,4,5,6} A={1,2,3} B={4,5} = B A φ
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17 Events have probability S={1,2,3,4,5,6} A={1,2,3} What is the probability of A? P(A) = 3/6 = 0.5
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18 Probability as a Relative Frequency (occurrence “in the long run”) Tossing a coin: The relative frequency of occurrences of an event A, should approach the probability P(A), as the number of trials grows (when the trials are random and independent of each other).
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19 Probability as ratio of sizes The probability of an event A: S A P in elements of number A in elements of number ) ( = Roulette example: A={1,13} } 100 , , 3 , 2 , 1 { = S 100 2 ) ( = A P
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Example: A letter is chosen at random from the word PROFIT. What is the probability that it is a vowel? S – {P,R,O,F,I,T}
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This note was uploaded on 11/03/2009 for the course IST IST taught by Professor N/a during the Spring '09 term at Anadolu University.

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web14 - Probability 1 Todays plan Probability Notations...

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