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HS 67BPS Chapter 101 Chapter 10 Introducing Probability

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HS 67BPS Chapter 102 Idea of Probability Probability is the science of chance behavior Chance behavior is unpredictable in the short run, but is predictable in the long run The probability of an event is its expected proportion in an infinite series of repetitions The probability of any outcome of a random variable is an expected (not observed) proportion

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HS 67BPS Chapter 103 How Probability Behaves Coin Toss Example Eventually, the proportion of heads approaches 0.5

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HS 67BPS Chapter 104 How Probability Behaves “Random number table example” The probability of a “0” in Table B is 1 in 10 (.10) Q: What proportion of the first 50 digits in Table B is a “0”? A: 3 of 50, or 0.06 Q: Shouldn’t it be 0.10? A: No. The run is too short to determine probability. (Probability is the proportion in an infinite series.)

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HS 67BPS Chapter 105 Probability models consist of two parts: 1)Sample Space (S) = the set of all possible outcomes of a random process. 2)Probabilities for each possible outcome in sample space S are listed. Probability Models Probability Model “toss a fair coin” S = {Head, Tail} Pr(heads) = 0.5 Pr(tails) = 0.5

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HS 67BPS Chapter 106 Rules of Probability

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HS 67BPS Chapter 107 Rule 1 (Possible Probabilities) Let A ≡ event A 0 ≤ Pr(A) ≤ 1 Probabilities are always between 0 and 1. Examples: Pr(A) = 0 means A never occurs Pr(A) = 1 means A always occurs Pr(A) =.25 means A occurs 25% of the time

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HS 67BPS Chapter 108 Rule 2 (Sample Space) Let S ≡ the entire Sample Space Pr(S) = 1 All probabilities in the sample space together must sum to 1 exactly. Example: Probability Model “toss a fair coin”, shows that Pr(heads) + Pr(tails) = 0.5 + 0.5 = 1.0

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HS 67BPS Chapter 109 Rule 3 (Complements) Let Ā ≡ the complement of event A Pr(Ā) = 1 – Pr(A) A complement of an event is its opposite For example: Let A ≡ survival then Ā ≡ death If Pr(A) = 0.95, then Pr(Ā) = 1 – 0.95 = 0.05

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HS 67BPS Chapter 1010 Events A and B are disjoint if they are mutually exclusive. When events are disjoint Pr(A or B) = Pr(A) + Pr(B) Age of mother at first birth (A) under 20: 25% (B) 20-24: 33% (C) 25+: 42% } Pr(B or C) = 33% + 42% = 75% Rule 4 (Disjoint events)

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HS 67BPS Chapter 1011 Discrete Random Variables Example: A couple wants three children. Let X ≡ the number of girls they will have This probability model is discrete: Discrete random variables address outcomes that take on only discrete (integer) values

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HS 67BPS Chapter 1012 Example Generate random number between 0 and 1 infinite possibilities. To assign probabilities for continuous random variables density models (recall Ch 3) Continuous Random Variables Continuous random variables form a continuum of possible outcomes. This is the density model for random numbers between 0 and 1

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HS 67BPS Chapter 1013 Area Under Curve (AUC) The AUC concept (Chapter 3) is essential to working with continuous random variables. Example: Select a number between 0 and 1 at random. Let X ≡ the random value. Pr(X <.5) =.5 Pr(X > 0.8) =.2

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HS 67BPS Chapter 1014 Normal Density Curves → ♀ Height X~N(64.5, 2.5) Z Scores Introduced in Ch 3: X~N(µ, ). Standardized Z~N(0, 1)

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HS 67BPS Chapter 1015 If I select a woman at random a 99.7% chance she is between 57" and 72 " 68-95-99.7 Rule Let X ≡ ♀ height (inches) X ~ N (64.5, 2.5) Use 68-95-99.7 rule to determine heights for 99.7% of ♀ μ ± 3σ = 64.5 ± 3(2.5) = 64.5 ± 7.5 = 57 to 72

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HS 67BPS Chapter 1016 Calculating Normal Probabilities when 68-95-99.7 rule does not apply Recall 4 step procedure (Ch 3) A: State B: Standardize C: Sketch D: Table A

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HS 67BPS Chapter 1017 Illustration: Normal Probabilities What is the probability a woman is between 68” and 70” tall? Recall X ~ N (64.5, 2.5) A: State: We are looking for Pr(68 < X < 70) B: Standardize Thus, Pr(68 < X < 70) = Pr(1.4 < Z < 2.2)

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HS 67BPS Chapter 1018 Illustration (cont.) C: Sketch D: Table A: Pr(1.4 < Z < 2.2) = Pr(Z < 2.2) − Pr(Z < 1.4) = 0.9861 − 0.9192 = 0.0669

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