5/23/11 Lecture 10 1 STOR 155 Introductory Statistics Lecture 10: Randomness and Probability Model The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL5/23/11 Lecture 10 2 Randomness & Probability • We call a phenomenon (or an experiment) random if individual outcomes are uncertain, but a regular distribution of outcomes emerges with a large number of repetitions. – Example: Toss a coin, gender of new born baby. • The probability of any outcome in a random experiment is approximately the proportion of times the outcome would occur in a very long series of independent repetitions, – using a long-term relative frequency to realize it … – In the early days, probability was associated with games of chance (gambling).5/23/11 Lecture 10 3 Probability as long term relative frequency --- two of many possible sample paths5/23/11 Lecture 10 4 Probability Model • Probability models attempt to model random behavior. • Consist of two parts: – A list of possible outcomes (sample space S) – An assignment of probabilities P to each outcome • The probability of an event A, denoted by P(A), can be considered as the long run relative frequency of the event A.5/23/11 Lecture 10 5 Sample Space and Events • Sample space S: the set of all possible outcomes in a random experiment. – Examples: Toss a coin. Record the side facing up. S ={{Heads}, {Tails} } = { H, T }. – Toss a coin twice. Record the side facing up each time. S = ? . – Toss a coin twice. Record the number of heads in the two tosses. S = ?. • Event: An outcome or a set of outcomes in a random experiment, – i.e. a subset of the sample space.5/23/11 Lecture 10 6 Sample Space a sample space of a random experiment is the set of all possible outcomes. Simple events The individual outcomes are called simple events. Event An event is a collection of one or more simple events Sample Spaces & Events Our objective is to determine P(A), the probability that an event A will occur.5/23/11 Lecture 10 7 Toss a coin 3 times Sample space S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. • There are 8 simple events, among which are E1 = {HHH} and E8 ={TTT}. • Some compound events include A = {at least two heads} = {HHH, HHT,HTH, THH} . B = {exactly two tails} = ?.5/23/11 Lecture 10 8 Boy or girl? • An experiment in a hospital consists of recording the gender of each newborn infant until the birth of a male is observed. • The sample space of this experiment is S = {M, FM, FFM, FFFM, ...} • The sample space contains an infinite number of outcomes.5/23/11 Lecture 10 9 Basic Concepts • The complement of an event A – the set of all outcomes in S that are not in A. – { not A } • The union of two events A and B – the event consisting of all outcomes that are either in A or in B or in both. – A B • The intersection of two events A and B – the event consisting of all outcomes that are in both events. – A B • When two evens A and B have no outcomes in common, they are said to be disjoint (or mutually exclusive) events.5/23/11 Lecture 10 10 Venn Diagram5/23/11 Lecture 10 11 Probability Rules • For any event A, 0 P(A) 1. • P(S) = 1. • If A and B are disjoint events, then P(A B) = P(A) + P(B). (addition rule for disjoint events) • For any event A, P( not A ) = 1 - P(A). (complement rule) • For any two events A and B, P(A B) = P(A) + P(B) - P(A B). (general addition rule) • If A and B are disjoint, then P(A B) = 0.5/23/11 Lecture 10 12 Equally Likely Outcomes • If there are k equally likely outcomes, then the probability assigned to each outcome is 1/k. P(A) = (# of outcomes in A) / k • Key: smart counting --- ``no omission, no duplication’’Birthday problem • In a group of 5 people, what is the chance that at least two of them share the same birthday? • Assume 365 days in one year, then the chance is 1 – p … Detailed work will be shown on the board … 5/23/11 Lecture 10 135/23/11 Lecture 10 14 • The label facing up, when a fair die is rolled, is observed. • Sample Space: S = { 1, 2, 3, 4, 5, 6}. • Every outcome is equally likely to occur. P(1) = P(2) = … = P(6) = 1/6. 1 2 4 5 6 3 Venn Diagram Roll a fair die once5/23/11 Lecture 10 15 • Consider the following events – A: The label observed is at most 2. – B: The label observed is an even number. – C: Label 4 turns up. • Find • P(A) • P( not A) • P(A and B) • P(A or C) • P(A or B) Roll a fair die once5/23/11 Lecture 10 16 Cards A card is drawn from an ordinary deck of 52 playing cards. What is the probability that the card is -- a club? -- a king? -- a club and a king? -- a club or a king? -- neither a club nor a king?5/23/11 Lecture 10 17 Glasses • In a group of 88 people in STOR 155, 11 out of 50 women and 8 out of 38 men wear glasses. • What is the probability that a person chosen at random from the group is a woman or someone who wears glasses?5/23/11 Lecture 10 18 Venn diagram with 3 events • A = {Google stock moves up today} • B = {Walmart stock moves up today} • C = {Exxon stock moves up today} P(A) = 0.1, P(B) = 0.2, P(C) = 0.5 P(A B) = 0.05, P(A C) = 0.04, P(B C) = 0.02 P(A B C) = 0.01 Find: (i) P( at least one of the 3 stocks go up) = (ii) P( both Google and Exxon go down) = (iii) P( only one of the 3 sticks goes up) =5/23/11 Lecture 10 19 Continued How to complete a Venn diagram? --- Insert a probability in each disjoint part --- ``inside-out’’ --- See details on the board …5/23/11 Lecture 10 20 Take Home Message • sample space, outcome, event • union (or), intersection (and), complement (not), disjoint • Venn diagram • Basic rules: – For any event A, P( not A) = 1 - P(A). – If A and B are disjoint, then P(A B) = 0. – For any two events A and B, P(A B) = P(A) + P(B) - P(A
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