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Chapter 15 – Probability Rules

Chapter 15 – Probability Rules. General Addition Rule. A and B are disjoint: P(A or B) = P(A) + P(B) When A and B are not disjoint, events may get double-counted P(A or B) = P(A) + P(B) – P(A and B) P(face card or club) P(pick a # 1-10 odd or mult of 3)

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Chapter 15 – Probability Rules

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  1. Chapter 15 – Probability Rules

  2. General Addition Rule A and B are disjoint: P(A or B) = P(A) + P(B) When A and B are not disjoint, events may get double-counted P(A or B) = P(A) + P(B) – P(A and B) P(face card or club) P(pick a # 1-10 odd or mult of 3) P(student is female or full-time)

  3. Example • 70% of students in a certain class are full-time, 40% are male, and 18% are both. • What is the probability that a student chosen at random is: • Part-time? • Female? • Full-time or male but not both? • Female and full-time?

  4. More Probability Below are the results of a survey of 478 students (grades 4, 5, 6) asking about their primary goals. If you select a student at random from this study, find: P(girl) P(girl and sports) P(sports) Table from Intro Stats, De Veaux

  5. Conditional Probability P(sports) = 90/478 = 0.188 P(sports | girl) = P(sports | boy) = P(girl | sports) = P(boy | sports) =

  6. Conditional Probability Formula From previous example with 70% full-time, 40% male and 18% both: P(male | full-time) = P(Full-time | male) =

  7. General Multiplication Rule A and B are independent: P(A and B) = P(A) x P(B) When A and B are dependent, P(B) affected by whether or not A occurs: P(A and B) = P(A) x P(B | A) or P(B) x P(A | B)

  8. Examples P(red card followed by another red card - without replacement) P(choosing a female from the class and then a male –without replacement) Jar of 25 quarters, 15 dimes, 10 nickels Draw 3 coins without replacement. P(all 3 quarters) P(none of the 3 are quarters) P(quarter and then 2 dimes) P(at least one of the 3 is a quarter)

  9. Testing for Independence Independence occurs when P(B | A) = P(B) P(grades | girl) P(sports | girl) P(grades) P(sports)

  10. Conditional Probability using a table Previous example with 70% full-time, 40% male and 18% both

  11. Tree Diagrams Results of study by the Harvard School of Public Health (“Binge Drinking on Campus: Results of a National Study”) 44% binge, 37% moderate, 19% abstain Results of study in American Journal of Health Behavior 17% binge drinkers in alcohol-related accident, 9% non-binge drinkers in alcohol-related accident Figure from Intro Stats, De Veaux

  12. Example Find the probabilities: P(R and favor) P(R) P(favor) P(R | favor) P(favor | R) Are party affiliation and position on death penalty independent? Table from Intro Stats, De Veaux

  13. Example Suppose a student at random is selected, calculate the following probabilities: P(Agriculture) P(Agriculture and firstborn) P(Agriculture or firstborn) P(Agriculture | firstborn) Table from Intro Stats, De Veaux

  14. Example • Suppose that 20% of the cars on campus are white. • If you select 4 cars at random, find the probability: • All 4 are white • None of the 4 are white • The first 2 are white and the last 2 aren’t • At least one of the 4 is white

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