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1STAT 13, UCLA, Ivo DinovSlide 1UCLA STAT 13Introduction toStatistical Methods for the Life and Health Sciences!Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology!Teaching Assistants: Sovia Lau, Jason ChengUCLA StatisticsUniversity of California, Los Angeles, Fall 2003http://www.stat.ucla.edu/~dinov/courses_students.htmlSTAT 13, UCLA, Ivo DinovSlide 2Chapter 4: Probabilities and Proportions!Where do probabilities come from?!Simple probability models!probability rules!Conditional probability!Statistical independenceSTAT 13, UCLA, Ivo DinovSlide 3Let's Make a Deal Paradox –aka, Monty Hall 3-door problem! This paradox is related to a popular television show in the 1970's. In the show, a contestant was given a choice of three doors/cards of which one contained a prize (diamond). The other two doors contained gag gifts like a chicken or a donkey (clubs). STAT 13, UCLA, Ivo DinovSlide 4Let's Make a Deal Paradox.! After the contestant chose an initial door, the host of the show then revealed an empty door among the two unchosen doors, and asks the contestant if he or she would like to switch to the other unchosen door. The question is should the contestant switch. Do the odds of winning increase by switching to the remaining door? 1.PickOnecard2.Show oneClub Card3. Change 1stpick?STAT 13, UCLA, Ivo DinovSlide 5Let's Make a Deal Paradox.! The intuition of most people tells them that each of the doors, the chosen door and the unchosen door, are equally likely to contain the prize so that there is a 50-50 chance of winning with either selection? This, however, is not the case. ! The probability of winning by using the switching technique is 2/3, while the odds of winning by not switching is 1/3. The easiest way to explain this is as follows:STAT 13, UCLA, Ivo DinovSlide 6Let's Make a Deal Paradox.! The probability of picking the wrong door in the initial stage of the game is 2/3. ! If the contestant picks the wrong door initially, the host must reveal the remaining empty door in the second stage of the game. Thus, if the contestant switches after picking the wrong door initially, the contestant will win the prize. ! The probability of winning by switching then reduces to the probability of picking the wrong door in the initial stage which is clearly 2/3. ! Demos:! file:///C:/Ivo.dir/UCLA_Classes/Applets.dir/SOCR/Prototype1.1/classes/TestExperiment.html! C:\Ivo.dir\UCLA_Classes\Applets.dir\StatGames.exe2STAT 13, UCLA, Ivo DinovSlide 70.40.50.6101102103104Number of tossesFigure 4.1.1 Proportion of heads versus number of tossesfor John Kerrich's coin tossing experiment.From Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000.Long run behavior of coin tossingSTAT 13, UCLA, Ivo DinovSlide 8Definitions …! The law of averages about the behavior of coin tosses –the relative proportion (relative frequency) of heads-to-tails in a coin toss experiment becomes more and more stableas the number of tosses increases. The law of averages applies to relative frequencies not absolute counts of #H and #T.! Two widely held misconceptions about what the law of averages about coin tosses:" Differences between the actual numbers of heads & tails becomes more and more variable with increase of the number of tosses – a seq. of 10 heads doesn’t increase the chance of a tail on the next trial." Coin toss results are fair, but behavior is still unpredictable.STAT 13, UCLA, Ivo DinovSlide 9Coin Toss Models! Is the coin tossing model adequate for describing the sex order of children in families? " This is a rough model which is not exact. In most countries rates of B/G is different; form 48% … 52%, usually. Birth rates of boys in some places are higher than girls, however, female population seems to be about 51%." Independence, if a second child is born the chance it has the same gender (as the first child) is slightly bigger.STAT 13, UCLA, Ivo DinovSlide 10220120140160180200JanFebMarAprMayJunJulAugSepOctNovDecMonth of the yearFigure 4.3.1 Average lottery numbers by month.Replotted from data in Fienberg [1971].From Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000.Data from a “random” draw366 cylinders (for each day in the year) for theUS Vietnam war draft. The N-th drawn number,corresp. to one B-day, indicating order of drafting.So, people born later inthe year tend to havelower lottery numbersand a bigger chance ofactually being drafted.STAT 13, UCLA, Ivo DinovSlide 11Types of Probability! Probability models have two essential components (sample space, the space of all possible outcomes from an experiment; and a list of probabilities for each event in the sample space). Where do the outcomes and the probabilities come from?! Probabilities from models– say mathematical/physical description of the sample space and the chance of each event. Construct a fair die tossing game.! Probabilities from data – data observations determine our probability distribution. Say we toss a coin 100 times and the observed Head/Tail counts are used as probabilities.! Subjective Probabilities – combining data and psychological factors to design a reasonable probability table (e.g., gambling, stock market).STAT 13, UCLA, Ivo DinovSlide 12Sample Spaces and Probabilities! When the relative frequency of an event in the past is used to estimate the probability that it will occur in the future, what assumptionis being made? " The underlying process is stable over time;" Our relative frequencies must be taken from large numbers for us to have confidence in them as probabilities.! All statisticians agree about how probabilities are to be combined and manipulated (in math terms), however, not all agree what probabilities should be associated with for a particular real-world event.! When a weather forecaster says that there is a 70% chance of rain tomorrow, what do you think this statement means? (Based on our past knowledge, according to the barometric pressure, temperature, etc. of the conditions we expect tomorrow, 70% of the time it did rain under such conditions.)3STAT 13, UCLA, Ivo DinovSlide 13Sample spaces and events! A sample space, S, for a random experiment is the set of all possible outcomes of the experiment.! An event is a collection of outcomes.! An event occurs if any outcome making up that event occurs.STAT 13, UCLA, Ivo DinovSlide 14! The complement of an event A, denoted ,occurs if and only if A does not occur.AA A(a) Sample space


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