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Probability

Probability. Toolbox of Probability Rules. Event. An event is the result of an observation or experiment, or the description of some potential outcome. Denoted by uppercase letters: A, B, C, …. Examples: Events. A = Event student has four exams in one day.

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Probability

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  1. Probability Toolbox of Probability Rules

  2. Event • An event is the result of an observation or experiment, or the description of some potential outcome. • Denoted by uppercase letters: A, B, C, …

  3. Examples: Events • A = Event student has four exams in one day. • B = Event PSU football team wins national championship. • C = Event that a fraternity is raided next weekend. Notation: The probability that an event A will occur is denoted as P(A).

  4. Tool 1 • The complement of an event A, denoted AC, is “the event that A does not happen.” • P(AC) = 1 - P(A)

  5. Example: Tool 1 • Suppose 1% of population is alcoholic. • Let A = event person is alcoholic. • Then AC = event person is not alcoholic. • P(AC) = 1 - 0.01 = 0.99 • That is, 99% of population is not alcoholic.

  6. Prelude to Tool 2 • The intersection of two events A and B, denoted “A and B”, is “the event that both A and B happen.” • Two events are independent if the events do not influence each other. That is, if event A occurs, it does not affect chances of B occurring, and vice versa.

  7. Example for Prelude to Tool 2 • Let A = event student passes this course • Let B = event student gives blood today • The intersection of the events, “A and B”, is the event that the student passes this course and the student gives blood today. • Is it OK to assume that A and B are independent?

  8. Example for Prelude to Tool 2 • Let A = event student passes this course • Let B = event student tries to pass this course • The intersection of the events, “A and B”, is the event that the student passes this course and the student tries to pass this course. • Is it OK to assume that A and B are not independent, that is “dependent”?

  9. Tool 2 • If two events are independent, then P(A and B) = P(A)  P(B). • If P(A and B) =P(A)  P(B), then the two events A and B are independent.

  10. Example: Tool 2 • Let A = event randomly selected student owns bike. P(A) = 0.36 • Let B = event student has significant other. P(B) = 0.45 • Assuming bike ownership is independent of having SO: P(A and B) = 0.36 × 0.45 = 0.16 • 16% of students own bike and have SO.

  11. Example: Tool 2 • Let A = event student is male. P(A) = 0.50 • Let B = event student is sleep deprived. P(B) = 0.60 • A and B = student is sleep deprived and male. P(A and B) = 0.30 • P(A) × P(B) = 0.50 × 0.60 = 0.30 • P(A and B) = P(A) × P(B). So, being male and being sleep-deprived are independent.

  12. Prelude to Tool 3 • The union of two events A and B, denoted A or B, is “the event that either A happens or B happens, or both A and B happen.” • Two events that cannot happen at the same time are called mutually exclusive events.

  13. Example to Prelude to Tool 3 • Let A = event student is drunk. • Let B = event student is sober. • A or B = event student is either drunk or sober. • Are A and B mutually exclusive?

  14. Example to Prelude to Tool 3 • Let A = event student is drunk • Let B = event student is in love • A or B = event student is either drunk or in love • Are A and B mutually exclusive?

  15. Tool 3 • If two events are mutually exclusive, then P(A or B) = P(A) + P(B). • If two events are not mutually exclusive, then P(A or B) =P(A)+P(B)-P(A and B).

  16. Example: Tool 3 • Let A = randomly selected student has two blue eyes. P(A) = 0.32 • Let B = randomly selected student has two brown eyes. P(B) = 0.38 • P(A or B) = 0.32 + 0.38 = 0.70

  17. Example: Tool 3 • Let A = event randomly selected student drinks alcohol. P(A) = 0.75 • Let B = event student ever tried marijuana. P(B) = 0.38 • A and B = event student drinks alcohol and has tried marijuana. • P(A and B) = 0.37 • P(A or B) = 0.75 + 0.38 - 0.37 = 0.76

  18. Tool 4 • The conditional probability of event B given A has already occurred, denoted P(B|A), is the probability that B will occur given that A has already occurred. • P(B|A) = P(A and B)  P(A) • P(A|B) = P(A and B)  P(B)

  19. Example: Tool 4 • Let A = event student owns bike, and B = event student has a significant other. • P(B|A) is the probability that a student has a significant other “given” (or “if”) he/she owns a bike. • P(A|B) is the probability that a student owns a bike “given” he/she has a significant other.

  20. Example: Tool 4 • Let A = event randomly selected student owns bike. P(A) = 0.36 • Let B = event randomly selected student has significant other. P(B) = 0.45 • P(A and B) = 0.17 • P(B|A) = 0.17 ÷ 0.36 = 0.47 • P(A|B) = 0.17 ÷ 0.45 = 0.38

  21. Tool 5 • Alternative definition of independence: • two events are independent if and only if P(A|B) = P(A) and P(B|A) = P(B). • That is, if two events are independent, then P(A|B) = P(A) and P(B|A) = P(B). • And, if P(A|B) = P(A) and P(B|A) = P(B), then A and B are independent.

  22. Example: Tool 5 • Let A = event student is female • Let B = event student abstains from alcohol • P(A) = 0.50 and P(B) = 0.12 • P(A|B) = 0.50 and P(B|A) = 0.12 • Are events A and B independent?

  23. Example: Tool 5 • Let A = event student is female • Let B = event student dyed hair • P(A) = 0.50 and P(B) = 0.40 • P(A|B) = 0.65 and P(B|A) = 0.52 • Are events A and B independent?

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