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Unformatted text preview: STAT 211 Prof Parzen CHAPTER 2 PROBABILITY, CONDITIONAL PROBABILITY, BAYES Probability theory enables us to measure uncertainity, chance, likelihood. Probability theory has applications to ex plain and predict observations in every aspect of life: science, engineering, management, financial planning for retirement,. Statistics learns from data probability models that fit observed data (by computing probabilities that a model is true). We study probability theory because it is the basis for statistical applica tions. 2.0 COIN TOSSING D ecisions are often made by tossing a coin which has two possi ble outcomes; head (often denoted as 1 or success S or s) and tails (often denoted 0 or failure F or f). When observing the sex (gender) of a newly born baby: possible outcomes are male (of ten denoted 4) and female (often denoted 6) The outcome of a single coin toss (the gender of a new baby) is unpredictable. However one may be able to predict proportions (of a large number of observations with specified outcomes). DEEFINITION: Denote by Proportion [heads] the proportion of heads in a large number of coin tosses; it is a number (between 0 and 1). Experience leads us to believe proportion is reproduci ble approximately and what is predictable are intervals in which the observed proportion lies with high probability. The concept of probability of heads on a coin toss, denoted Prob [heads], is defined on order to provide a model to predict and/or explain the observed proportion of heads in a large number of coin tosses. PROBABILITY AND STATISTICAL INFERENCE: Prob ability makes assumptions about probability p of heads in a coin toss, and computes probabilities that future values of Proportion [heads] will be in a specified interval. Statistical inference ob 2 serves the value of Proportion [heads], in n tosses of a coin whose probability p is not assumed known, and determines probabilities that p is in a specified interval. Fair coin : A coin whose probability p of heads equals .5 is called a fair coin. We apply this concept in practice by expect ing a fair coin to fall heads approximately 5000 times in 10000 tosses (50 heads in 100 tosses), Proportion Heads in 10000 trials Fair Coin =5000 approximately Precise statements are not made about the exact number of heads. Rather we say with probability 95% a fair coin will fall heads in 10000 n = tosses a proportion between .49 and .51; in 100 n = tosses of a fair coin, the proportion of heads will be between .4 and .6 with probability 95%. Note that the interval of proportions we expect to observe changes as we change the sample size n , and converges to a sin gle number (the true value p) as the sample size tends to infinity....
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This note was uploaded on 03/27/2008 for the course STAT 211 taught by Professor Parzen during the Fall '07 term at Texas A&M.
 Fall '07
 Parzen
 Conditional Probability, Probability

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