5/24/10 Lecture 10 1STOR 155 Introductory StatisticsLecture 10: Randomness and Probability ModelThe UNIVERSITY of NORTH CAROLINAat CHAPEL HILL5/24/10 Lecture 10 2Randomness & 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).5/24/10 Lecture 10 3Probability as long term relative frequency--- two of many possible sample paths5/24/10 Lecture 10 4Probability 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.5/24/10 Lecture 10 5Sample 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.5/24/10 Lecture 10 6Sample Spacea sample space of a random experimentis the set of all possible outcomes.Simple eventsThe individual outcomes are called simple events. EventAn event is a collectionof one or more simple eventsSample Spaces & EventsOur objective is to determine P(A), the probability that anevent A will occur.5/24/10 Lecture 10 7Toss a coin 3 timesSample 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 includeA = {at least two heads} = {HHH, HHT,HTH, THH} . B = {exactly two tails} = ?.5/24/10 Lecture 10 8Boy 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 isS = {M, FM, FFM, FFFM, ...}• The sample space contains an infinitenumber of outcomes.5/24/10 Lecture 10 9Basic 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.5/24/10 Lecture 10 10Venn Diagram5/24/10 Lecture 10 11Probability Rules• For any event A, 0 P(A) 1.• P(S) = 1. • If A and B are disjoint events, thenP(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) = 0.5/24/10 Lecture 10 12Equally 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…5/24/10 Lecture 10 135/24/10 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) = 1/6.124563Venn DiagramRoll a fair die once5/24/10 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 once5/24/10 Lecture 10 16CardsA 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?5/24/10 Lecture 10 17Glasses• 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?5/24/10 Lecture 10 18Venn 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) = 0.5P(A B) = 0.05, P(A C) = 0.04, P(B C) = 0.02P(A B C) = 0.01Find: (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) =5/24/10 Lecture 10 19ContinuedHow to complete a Venn diagram?--- Insert a probability in each disjoint part--- ``inside-out’’--- See details on the board …5/24/10 Lecture 10 20Take 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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