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Chapter 10 Introducing Probability HS 67 BPS Chapter 10 1 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 HS 67 The probability of any outcome of a random variable is an expected (not observed) proportion BPS Chapter 10 2 How Probability Behaves Coin Toss Example Eventually, the proportion of heads approaches 0.5 HS 67 BPS Chapter 10 3 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.) HS 67 BPS Chapter 10 4 Probability Models 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 Model “toss a fair coin” S = {Head, Tail} Pr(heads) = 0.5 Pr(tails) = 0.5 HS 67 BPS Chapter 10 5 Rules of Probability HS 67 BPS Chapter 10 6 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 HS 67 BPS Chapter 10 7 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 HS 67 BPS Chapter 10 8 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 HS 67 BPS Chapter 10 9 Rule 4 (Disjoint events) 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% Pr(B or C) = 33% + 42% = 75% (C) 25+: 42% } HS 67 BPS Chapter 10 10 Discrete Random Variables Discrete random variables address outcomes that take on only discrete (integer) values Example: A couple wants three children. Let X ≡ the number of girls they will have This probability model is discrete: HS 67 BPS Chapter 10 11 Continuous Random Variables Continuous random variables form a continuum of possible outcomes. • Example Generate random number between 0 and 1 infinite possibilities. • To assign probabilities for continuous random variables density models (recall Ch 3) HS 67 This is the density model for random numbers between 0 and 1 BPS Chapter 10 12 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 HS 67 BPS Chapter 10 13 Normal Density Curves Introduced in Ch 3: X~N(µ, ). ♀ Height X~N(64.5, 2.5) → z x Standardized Z~N(0, 1) z x Z Scores HS 67 BPS Chapter 10 14 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 HS 67 If I select a woman at random a 99.7% chance she is between 57" and 72" BPS Chapter 10 15 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 HS 67 BPS Chapter 10 16 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 (68 64.5) z 1.4 2.5 (70 64.5) z 2.2 2.5 Thus, Pr(68 < X < 70) = Pr(1.4 < Z < 2.2) HS 67 BPS Chapter 10 17 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 HS 67 BPS Chapter 10 18