1 / 18

Chapter 10

Chapter 10. Introducing Probability . The probability of any outcome of a random variable is an expected (not observed) proportion. 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

zohar
Download Presentation

Chapter 10

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 10 Introducing Probability BPS Chapter 10

  2. The probability of any outcome of a random variable is an expected (not observed) proportion 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 BPS Chapter 10

  3. How Probability BehavesCoin Toss Example Eventually, the proportion of heads approaches 0.5 BPS Chapter 10

  4. 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.) BPS Chapter 10

  5. Probability Models • Probability models consist of two parts: • Sample Space (S) = the set of all possible outcomes of a random process. • 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 BPS Chapter 10

  6. Rules of Probability BPS Chapter 10

  7. 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 BPS Chapter 10

  8. 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 BPS Chapter 10

  9. 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 BPS Chapter 10

  10. 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% (C) 25+: 42% } Pr(B or C) = 33% + 42% = 75% BPS Chapter 10

  11. 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: BPS Chapter 10

  12. This is the density model for random numbers between 0 and 1 Continuous Random Variables Continuousrandom 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) BPS Chapter 10

  13. 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 BPS Chapter 10

  14. Normal Density Curves Introduced in Ch 3: X~N(µ, ). ♀ Height X~N(64.5, 2.5) → Standardized Z~N(0, 1) Z Scores BPS Chapter 10

  15. 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 If I select a woman at random  a 99.7% chance she is between 57" and 72" BPS Chapter 10

  16. 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 BPS Chapter 10

  17. 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) BPS Chapter 10

  18. 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 BPS Chapter 10

More Related