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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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