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# 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
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
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
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
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
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}
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
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
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).
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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web14 - Probability 1 Todays plan Probability Notations...

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