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Probability

STAT 101 Dr. Kari Lock Morgan. Probability. SECTION 11.1 Events and, or, if Disjoint Independent Law of total probability. Confounding Variables. Once GDP is accounted for, electricity use is no longer a significant predictor of life expectancy.

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Probability

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  1. STAT 101 Dr. Kari Lock Morgan Probability • SECTION 11.1 • Events • and, or, if • Disjoint • Independent • Law of total probability

  2. Confounding Variables Once GDP is accounted for, electricity use is no longer a significant predictor of life expectancy. Once GDP is accounted for, electricity use is no longer a significant predictor of life expectancy. Even after accounting for GDP, cell phone subscriptions per capita is still a significant predictor of life expectancy.

  3. Confounding Variables (review) • Multiple regression is one potential way to account for confounding variables • This is most commonly used in practice across a wide variety of fields, but is quite sensitive to the conditions for the linear model (particularly linearity) • You can only account for confounding variables that you have data on, so it is still very hard to make true causal conclusions without a randomized experiment

  4. Event • An event is something that either happens or doesn’t happen, or something that either is true or is not true • Examples: • A randomly selected card is a Heart • The response variable Y > 90 • A randomly selected person is male • It rains today

  5. Probability • The probabilityof event A, P(A), is the probability that A will happen • Probability is always between 0 and 1 • Probability always refers to an event • P(A) = 1 means A will definitely happen • P(A) = 0 means A will definitely not happen

  6. Probability Examples • Y = number of siblings. P(Y = 1) = 0.481 • (based on survey data) • Y: final grade in STAT 101. P(Y > 90) = 0.338 • (based on last year’s class) • P(Gender = male) = 0.506 • (for Duke students, www.usnews.com) • P(it rains today) = 0.3 • (www.weather.com)

  7. Sexual Orientation • What are the sexual orientation demographics of American adults? • We need data! • Data collected in 2009 on a random sample of American adults (National Survey of Sexual Health and Behavior)

  8. Sexual Orientation Herbenick D, Reece M, Schick V, Sanders SA, Dodge B, and Fortenberry JD (2010). Sexual behavior in the United States: Results from a national probability sample of men and women ages 14–94. Journal of Sexual Medicine;7(suppl 5):255–265.

  9. Sexual Orientation What is the probability that an American adult is homosexual? 128/5042 = 0.025 128/4673 = 0.027 105/2521 = 0.04 I got a different answer

  10. Two Events • P(A and B)is the probability that both events A and B will happen • P(A or B) is the probability that either event A or event B will happen

  11. Two Events A or B: all color B A and B A

  12. Venn Diagram

  13. Venn Diagram

  14. Sexual Orientation What is the probability that an American adult is male and homosexual? 105/128 = 0.82 105/2521 = 0.04 105/5042 = 0.021 I got a different answer

  15. Sexual Orientation What is the probability that an American adult is female or bisexual? 2679/5042 = 0.531 2587/5042 = 0.513 92/2521 = 0.036 I got a different answer

  16. P(A or B) B A

  17. Sexual Orientation What is the probability that an American adult is not heterosexual? 369/5042 = 0.073 2587/5042 = 0.513 92/2521 = 0.036 I got a different answer

  18. P(not A)

  19. Caffeine • Based on last year’s survey data, 52% of students drink caffeine in the morning, 48% of students drink caffeine in the afternoon, and 37% drink caffeine in the morning and the afternoon. What percent of students do not drink caffeine in the morning or the afternoon? • 63% • 37% • 100% • 50% P( not(morning or afternoon) = 1 – P(morning or afternoon) = 1 – [P(morning)+P(afternoon) – P(morning and afternoon)] = 1 – [0.52 + 0.48 – 0.37] = 1 – 0.63 = 0.37

  20. Conditional Probability • P(A if B)is the probability of A, if we know B has happened • This is read in multiple ways: • “probability of A if B” • “probability of A given B” • “probability of A conditional on B” • You may also see this written as P(A | B)

  21. Sexual Orientation What is the probability that an American adult male is homosexual? 105/128 = 0.82 105/2521 = 0.04 105/5042 = 0.021 I got a different answer P(homosexual if male)

  22. Sexual Orientation What is the probability that an American adult homosexual is male? 105/128 = 0.82 105/2521 = 0.04 105/5042 = 0.021 I got a different answer P(male if homosexual)

  23. Conditional Probability P(homosexual if male) = 0.04 P(male if homosexual) = 0.82

  24. Conditional Probability B A

  25. Caffeine Based on last year’s survey data, 52% of students drink caffeine in the morning, 48% of students drink caffeine in the afternoon, and 37% drink caffeine in the morning and the afternoon. What percent of students who drink caffeine in the morning also drink caffeine in the afternoon? 77% 37% 71% P( afternoon if morning) = P(afternoon and morning)/P(morning) = 0.37/0.52 = 0.71

  26. Helpful Tip If the table problems are easier for your than the sentence problems, try to first convert what you know into a table. 52% of students drink caffeine in the morning, 48% of students drink caffeine in the afternoon, and 37% drink caffeine in the morning and the afternoon 37 15 52 37 11 48 52 100 48 P( afternoon if morning) = 37/52 = 0.71

  27. P(A and B)

  28. Duke Rank and Experience 60% of STAT 101 students rank their Duke experience as “Excellent,” and Duke was the first choice school for 59% of those who ranked their Duke experience as excellent. What percentage of STAT 101 students had Duke as a first choice and rank their experience here as excellent? 60% 59% 35% 41% P( first choice and excellent) = P(first choice if excellent)P(excellent) = 0.59  0.60 = 0.354

  29. Summary

  30. Disjoint Events • Events A and B are disjoint or mutually exclusiveif only one of the two events can happen • Think of two events that are disjoint, and two events that are not disjoint.

  31. Disjoint Events If A and B are disjoint, then P(A or B) = P(A) + P(B) P(A and B) = P(A)P(B) P(A or B) = P(A) + P(B) – P(A and B) If A and B are disjoint, then both cannot happen, so P(A and B) = 0.

  32. P(A or B) B B A A

  33. Independence • Events A and B are independent if P(A if B) = P(A). • Intuitively, knowing that event B happened does not change the probability that event A happened. • Think of two events that are independent, and two events that are not independent.

  34. Independent Events If A and B are independent, then P(A or B) = P(A) + P(B) P(A and B) = P(A)P(B) P( A and B) = P(A if B)P(B) If A and B are independent, then P(A if B) = P(A), so P(A and B) = P(A)P(B)

  35. P(A and B)

  36. Disjoint and Independent Assuming that P(A) > 0 and P(B) > 0, then disjoint events are Independent Not independent Need more information to determine whether the events are also independent If A and B are disjoint, then A cannot happen if B has happened, so P(A if B) = 0. If P(A) > 0, then P(A if B) ≠ P(A) so A and B are not independent.

  37. To Do • Read 11.1 • Read The Bayesian Heresyfor next class • Do Project 9 (due Wednesday, 4/23) • Do Homework 9 (due Wednesday, 4/23)

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