Venn Diagrams; Probability Laws. Notes. Set Operations and Relations. Venn Diagram 2.1. Venn Diagrams; Probability Laws. Notes

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1 Lecture 2 s; Text: A Course in Probability by Weiss 2.4 STAT 225 Introduction to Probability Models January 8, 2014 s; Whitney Huang Purdue University 2.1 Agenda s;

2 Intersection: the intersection of two sets A and B, denoted by (A B), is the set that contains all elements of A that also belong to B AND s; Example: Let A = {1, 2, 3} and B = {1, 2, 4, 5}, then A B = {1, 2} Union: the union of two sets A and B, denoted by (A B), is the set of all elements that belong to either A or B OR Example: Let A = {1, 2, 3} and B = {1, 2, 4, 5}, then A B = {1, 2, 3, 4, 5} 2.3 Example 8 Refer to Example 2, where we flipped 3 fair coins. Let A be the event of exactly 2 tails. Let B be the event that the first 2 tosses are tails. Let C be the event that all 3 tosses are tails. What are A B, A C, and (A B) C? Solution. s; 2.4

3 Logical Relationships among Sets Mutually exclusive: refers to two (or more) events that cannot both occur when the random experiment is formed. A B = Exhaustive: refers to event(s) that comprise the sample space. A B = Ω s; Partition: events that are both mutually exclusive and exhaustive. A B = and A B = Ω 2.5 Complement Rule The complement rule is a way to calculate a probability based on the probability of its complement. 1 By the definition of complement s; A A c = Ω 2 Apply the probability operator P(A A c ) = P(Ω) = 1 3 Since A and A c are mutually exclusive P(A A c ) = P(A) + P(A c ) 4 Hence we get P(A) = 1 P(A c ) 2.6

4 Example 9 Suppose we rolled a fair, six sided die 10 times. Let T be the event that we roll at least 1 three. If one were to calculate T you would need to find the probability of 1 three, 2 threes,, and 10 threes and add them all up. However, you can use the complement rule to calculate P(T ) s; Solution. 2.7 A Venn diagram is a diagram that shows all possible logical relations between a finite collection of sets. s; 2.8

5 General Addition Rule The general addition rule is a way of finding the probability of a union of 2 events. It is P(A B) = P(A) + P(B) P(A B) s; 2.9 Inclusion Exclusion Principle The inclusion exclusion principle is a way to extend the general addition rule to 3 or more events. Here we will limit it to 3 events. P(A B C) = P(A) + P(B) + P(C) P(A B) P(B C) P(A C) + P(A B C) s; 2.10

6 Law of Partitions The law of partitions is a way to calculate the probability of an event. Let A 1, A 2,, A k form a partition of Ω. Then, for all events B, s; P(B) = k P(A i B) i= DeMorgan s Law Let A and B be subsets of Ω. Then (A B) c = A c B c (A B) c = A c B c s; 2.12

7 Example 10 Three of the major commercial computer operating systems are Windows, Mac OS, and Red Hat Linux Enterprise. A Computer Science professor selects 50 of her students and asks which of these three operating systems they use. The results for the 50 students are summarized below. 30 students use Windows #(W ) = students use at least two of the operating systems #(W M W L M L) = 16 9 students use all three operating systems #(W M L) = 9 18 students use Mac OS #(M) = students use at least one of the operating systems #(W M L) = students use both Windows and Linux #(W L) = students use both Windows and Mac OS #(W M) = 11 s; 2.13 Example 10: Venn diagram Use the above information to complete a three-way Venn diagram. s; 2.14

8 Example 10 (cont d) Using the Venn diagram summarizing the distribution of operating system use previously described, Let W denotes Windows, M denotes Mac OS, and L denotes Red Hat Linux Enterprise. Calculate the following: 1 #(W c M c ) 2 P(W c M c ) 3 #(W M L) s; Solution More Laws Let A, B, and C be subsets of Ω Distributive law A (B C) = (A B) (A C) A (B C) = (A B) (A C) Associative law A B = B A A B = B A s; Commutative law A (B C) = (A B) C A (B C) = (A B) C 2.16

9 Summary In this lecture, we learned Set operations: union, intersection Set relations: mutually exclusive, Exhaustive, Partition Venn diagram: a handy tool to visualize set relationships Probability laws: general addition rule, complement rule s; 2.17

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