Introduction, Sample Space, EventsJuana [email protected] Department of StatisticsJ. Sanchez Introduction, Sample Space, EventsAnnouncementsPlease, visit CCLE daily ccle.ucla.edu to stay current on what isbeing done each week, access homework, practice exams, etc.. If itis posted, it is assumed known by you. You are responsible.Announcements are made in lecture and posted in CCLE.J. Sanchez Introduction, Sample Space, EventsOutlineI. Course syllabus summaryII. Introduction to Probability. Sample Spaces.III. Sample SpaceIV. EventJ. Sanchez Introduction, Sample Space, EventsI. Course SyllabusCourse syllabus reading is required.A copy of the syllabus has been handed out today. Pick up one fromthe front desk if you did not get one.The syllabus is also posted in CCLE in the syllabus folder.When you have an administrative question, read first the syllabus tosee if it is answered there.We will go over the main things in the syllabus separately now,particularly exam dates. Read more during the weekend and we willreview more syllabus next class.J. Sanchez Introduction, Sample Space, EventsII. Introduction to Probability. Sample Spaces.Throughout the centuries, most serious gamblers have developed afirm grasp of the relative chances of drawing the various hands atcards and of the various outcomes of throwing dice, simply by beingintensely involved and observing many hundreds of games (Empiricalprobabilities).But slowly, gamblers realized that games of chance might becomemore profitable if they could involve the mathematicians of the dayin calculating the odds involved. That gave rise to the mathematicalmodels for chance phenomena.This course covers the empirical approach (by practicing, or with thehelp of computers via simulation) and the mathematical approach,with applications of probability. We apply to a wide variety ofphenomena: genetics, inventory, actuarial science, military, kinetictheory, electricity, computer science, medicine, etc.J. Sanchez Introduction, Sample Space, EventsExperimental approachLaws of chance can be observed through experimentation.For example, we can show mathematically that the probability thatat least two people in a group of 10 people share a birthmonth is1 −12!2!1210. But we can study the same thing by repeating manytimes an experiment that consists of rolling a twelve-sided die 10times and recording whether there are any numbers repeated or not.The mathematical solution is the probability. The result of therepetition of experiment is the experimental probability or estimate.Experimentation is so intertwined with the theory of probability thatit is customary to call all situations in which outcomes are notpredictable with certainty as experiments.J. Sanchez Introduction, Sample Space, EventsSimulationFigure : Simulating dice rolls in RJ. Sanchez Introduction, Sample Space, Events”Chance is accepted as a fundamental feature of the universe, asphysicists have asserted the probabilistic nature of matter at the particlelevel. Basic postulate of quantum theory, the fundamental theory ofmatter is that individual events at the subatomic level are notreproducible at will or by experiment, not predictable in theory. ” MaxBorn.ProbabilityProbability theory is concerned with situations in which the outcomes arerandom (not predictable with certainty). Generically, such situations arecalled experiments.J. Sanchez Introduction, Sample Space, EventsIII. Sample SpaceWhat is the sample space?The set of all possible outcomes of an experiment is called the SampleSpace. Probability is defined on the events in a sample space. We denotethe Sample Space by ”S”. The first endeavor in any probabilitycalculation is to determine the Sample Space.ExampleExperiment is tossing three coinsS = {HHH , HHT , HTH, HTT , THH, THT , TTH, TTT }where outcome HHH occurs if the three coins are head.Figure : Sometimes a tree helps figure out the outcomes in the sample SpaceJ. Sanchez Introduction, Sample Space, EventsExampleExperiment is observing the length of time between successiveearthquakes.S = {t | t ≥ 0} The outcome ”takes 3 hours” is said to occur ift = 180minutes, 0seconds.ExampleExperiment is observing SAT scores for a student randomly chosenamong those that have taken the SAT. Note: SAT is a standardized testfor College Admissions. Scores are multiples of 10. There are 3 sections,reading, math, writing, each between 200 and 800. So the total score isbetween 600 to 2400.S = {x | 600 ≤ x ≤ 2400 such that x is multiple of 10}J. Sanchez Introduction, Sample Space, EventsExampleThe experiment is observing the reaction times to a certain stimulus, thenS = {x : 0 ≤ x ≤ ∞}ExampleThe experiment consists of screening individuals in a town until the onecarrying the virus H is foundS = {H , NH, NNH , NNNH, NNNNH , .....}J. Sanchez Introduction, Sample Space, EventsClassification of Sample SpacesCountable: can have finite number of elements or infinitely countable(elements can be put into 1-1 correspondence with integers)Uncountable: elements can’t be put into 1-1 correspondence withintegers; finite or infinite intervals of real numbers.J. Sanchez Introduction, Sample Space, EventsIV. EventsEventA collection of possible outcomes of an experiment that satisfy the samecondition, i.e., a subset of S. If the outcome of an experiment iscontained in the collection, then we say that the event has occurred. Thealgebra of set theory carries over directly into probability theory.ExampleA computer student can repeat an examination until it is passed, but it isallowed to attempt the examination at most four times. List the samplespace S and the event E that the student passes.Let P= passes and let Pc= Does not pass.S = {P , PcP, PcPcP, PcPcPcP, PcPcPcPc}E = {All outcomes in S except outcome (PcPcPcPc)}J. Sanchez Introduction, Sample Space, EventsPracticeDescribe events that could be defined in each of the following Samplespaces.ExampleExperiment is tossing three coinsExampleExperiment is observing the length of time between successiveearthquakes.ExampleExperiment is observing SAT scores for a student randomly chosenamong those that have taken the SAT.ExampleThe experiment is carrying out a database search for a particular item.J. Sanchez Introduction, Sample Space, EventsEvent OperationsUnionEvent consisting of elements of S that belong to either A or B or both(i.e., that are in at least one of the sets). It is written symbolically asA ∪ B .C = A ∪ B = {x : x ∈
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