lecture_4_1,2 - Probability and inference Randomness...

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    Probability and inference Randomness; Probability models IPS chapters 4.1 and 4.2 © 2006 W.H. Freeman and Company
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Objectives (IPS chapters 4.1 and 4.2) Randomness; Probability models Randomness and probability Probability models Probability rules Probabilities: finite number of outcomes Probabilities: equally likely outcomes
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A phenomenon is random if individual outcomes are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions. Randomness and probability The probability of any outcome of a random phenomenon can be defined as the proportion of times the outcome would occur in a very long series of repetitions.
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Coin toss First series of tosses Second series The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials.
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Probability models describe mathematically the outcome of a random phenomenon and consist of two parts: 1) S = Sample Space : This is a set, or list, of all possible outcomes of a random phenomenon. 2) A probability for each possible outcome in the sample space S . Probability models Example: Probability Model for a Coin Toss : S = {Head, Tail} Probability of heads = 0.5 Probability of tails = 0.5
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Example: Roll a Die Roll a (6-sided, fair) die. What are the possible outcomes? What are the probabilities of each of these possible outcomes?
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A. A basketball player shoots three free throws. What are the possible sequences of hits (H) and misses (M)? H H H - HHH M M M - HHM H - HMH M - HMM S = { HHH, HHM, HMH, HMM, MHH, MHM, MMH, MMM } Note: 8 elements, 2 3 B. A basketball player shoots three free throws. What are the possible numbers of hits? S = { 0, 1, 2, 3 } C. A nutrition researcher feeds a new diet to a young male white rat. What are the possible outcomes of weight gain/loss (in grams)? S = [any number] Sample spaces It’s the question that determines the sample space.
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Another example: M&M candies Color Brown Red Yellow Green Orange Blue Probability 0.3 0.2 0.2 0.1 0.1 0.1 In any bag of M&M candies, the candies have one of six colors. The proportion
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