Recap. Probability (section 1.1) The study of randomness and uncertainty Chances, odds, likelihood, expected, probably, on average,... PROBABILITY Population Sample INFERENTIAL STATISTICS Today. Formulation Sample spaces and events Set (Theory) Algebra Just some Shorthand notations for long sentences Why do we need this shorthand Probability measures are assigned to subsets of a set Operations on sets containment, equality, union, intersection, complementation DeMorgan s Law 1
Sample Spaces Outcome: a possible result of the random phenomenon studied. Experiment: any process leading to an uncertain outcome. (but it is known that the outcome will be one of several possible outcomes). The sample space of an experiment is the set of all possible outcomes. We will denote the sample space by S. Example: Toss a coin: S = { heads, tails } = { H, T } Observe Temperature at 7AM at Columbus airport. S? Roll a six-sided die: S = { 1,2,3,4,5,6 } Eamples of Sample Spaces Toss 2 coins Experiment 1: Outcome from tossing 2 coins (and assume they are distinguishable) S = { HH, HT, TH, TT} Experiment 2: # of heads from tossing 2 coins S = { 0, 1, 2} Roll 2 dice Experiment 3: Outcome from rolling 2 dice (and assume they are indistinguishable) S = { 11, 12, 13, 14, 15, 16, 22, 23, 24, 25, 26, 33, 34, 35, 36, 44, 45, 46, 55, 56, 66 } Experiment 4: Sum of the spots from rolling 2 dice S = { 2, 3, 4, 5,, 12 } 2
Events An event is a collection (subset) of outcomes in the sample space. Experiment S E The event E occurs if the outcome of an experiment is in E. Example: Throw a die: S = { 1, 2, 3, 4, 5, 6 } Throw produces an even #: E = { 2, 4, 6 } Throw produces a value 3: E = { 1, 2, 3 } Throw an even number 3: E = { 2 } Properties of Events Events E 1 and E 2 are mutually exclusive if no outcome is in both E 1 and E 2. E 1 E 2 E 1 E 2 = Φ Example: E 1 = throw even value with a single die E 2 = throw odd value with a single die E 1 = { 2, 4, 6 } E 2 = { 1, 3, 5 } E 1 E 2 = Φ 3
Example (Exercise 2.6 in book) A college library has five copies of a certain text on reserve. Two copies (1 and 2) are first printings, and the other three (3, 4, and 5) are second printings. A student examines these books in random order, stopping only when a second printing has been selected. One possible outcome is 5, and another is 213. a) List the outcomes in S S = { 3,4,5, 13,14,15, 23,24,25, 123,124,125, 213,214,215 } b) Let A denote the event that exactly one book must be examined. What outcomes are in A? A = { 3,4,5 } c) Let B be the event that book 5 is the one selected. What outcomes are in B? B = { 5, 15, 25, 125, 215 } d) Let C be the event that book 1 is not examined. What outcomes are in C? C = { 3,4,5, 23,24,25 } Useful Set (Theory) Algebra Notations A set is a well-defined group of elements/members. We denote sets with CAPITAL letters. We will call these elements elementary outcomes. A = all dates on which Stat 427 meets in SP 2011 = { 3/28, 3/30, 4/1, 4/4,, 6/3} 1. Containment (subset) The set A is said to be contained in B when all outcomes of A are in B. A = { 2,4,6,8 }, B = { 1,2,3,, 9, 10 } A = all dates on which Stat 427 meets in SP 2010 B = all days (all working days) in SP 2010 quarter, 2. Equality Two sets A, B are said to be equal when they contain the same outcomes. A = { 1,2,3 }, B = { 2,3,1 } then A = B (order does not matter) A = B if and only if C = { 1,1,2,3,3,3 }, then A = B = C (duplicates does not matter) A B & A B A B A B 4
Basic Set Operations 1. Union (combination) A union of two sets A and B (denoted by A U B) is the set consisting of all outcomes in A, in B, or in both (i.e, elements in at least one of the events). A = { 1,2,3 }, B = { 2,3,4,6 } then A U B = { 1,2,3,4,6 } 2. Intersection (common) A intersection of two sets A and B (denoted by A B) is the set consisting of all outcomes that are in both A & B. A = { 1,2,3 }, B = { 2,4,6 } A B then A B = { 2 } 3. Complement (negation) The complement of a set A (denoted by A or A c ) is the event consisting of all outcomes in the original domain (sample space) that are not contained in A. S = { 1,2,3,4,5,6 }, A = { 1,3,5 } then A =A c = { 2,4,6 } A U B A c 1. 2. Fundamental Relationships (DeMorgan s Law) A = all OSU students in Stat 427 B = all OSU students majoring in Mech. Eng. (A U B) c = (either in Stat 427 or majoring in Mech. Eng) c = neither in Stat 427 nor majoring in Mech. Eng = (in Stat 427) c and (majoring in Mech. Eng) c = A c B c (A B) c = (both in Stat 427 and majoring in Mech. Eng) c = not in both at the same time = either not in Stat 427 or not majoring in Mech. Eng = (in Stat 427) c or (majoring in Mech. Eng) c = A c U B c 5
Preview. Axioms and Properties of Probability Axiom: 1) P(S) = 1 2) P(E) 0 for every event E S 3) For disjoint events, Properties: 1) P( ) = 0 2) 3) 4) 6