Chapter Five Elementary Probability Theory

1. Understandable StatisticsSeventh EditionBy Brase and BrasePrepared by: Lynn SmithGloucester County College Chapter Five Elementary Probability Theory

2. Probability • Probability is a numerical measurement of likelihood of an event. • The probability of any event is a number between zero and one. • Events with probability close to one are more likely to occur. • If an event has probability equal to one, the event is certain to occur.

3. Probability Notation If A represents an event, P(A) represents the probability of A.

4. Three methods to find probabilities: • Intuition (subjective) • Relative frequency (experimental) • Equally likely outcomes (theoretical)

5. Intuition method(subjective; based on opinion or experience) based upon our level of confidence in the result Example: I am 95% sure that I will attend the party.

6. Probability as Relative Frequency(experimental; what has just happened) Probability of an event = the fraction of the time that the event occurred in the past = f n where f = frequency of an event n = sample size

7. Example of Probability as Relative Frequency If you note that 57 of the last 100 applicants for a job have been female, the probability that the next applicant is female would be:

8. Law of Large Numbers In the long run, as the sample size increases and increases, the relative frequencies of outcomes get closer and closer to the theoretical (or equally likely) probability value.

9. Equally likely outcomes(theoretical; if all results have the same chance) No one result is expected to occur more frequently than any other.

10. Three methods to find probabilities: • Intuition (subjective) • Relative frequency (experimental) • Equally likely outcomes (theoretical)

11. Probability of an event when outcomes are equally likely =

12. Example of Equally Likely Outcome Method When rolling a die, the probability of getting a number less than three =

13. Statistical Experiment activity that results in a definite outcome

14. Sample Space set of all possible outcomes of an experiment

15. Sample Space for the rolling of an ordinary die: 1, 2, 3, 4, 5, 6

16. Three methods to find probabilities: • Intuition (subjective) • Relative frequency (experimental) • Equally likely outcomes (theoretical)

17. For the experiment of rolling an ordinary die: 3 = 1 6 2 • P(even number) = • P(result less than four) = • P(not getting a two) = 3 = 1 6 2 5 6

18. Complement of Event A the event not A

19. Probability of a Complement P(not A) = 1 – P(A)

20. Probability of a Complement If the probability that it will snow today is 30%, P(It will not snow) = 1 – P(snow) = 1 – 0.30 = 0.70

21. Formal Probability (cont.) Complement Rule: • The set of outcomes that are not in the event A is called the complement of A, denoted AC. • The probability of an event occurring is 1 minus the probability that it doesn’t occur: P(A) = 1 – P(AC)

22. Probabilities of an Event and its Complement • Denote the probability of an event by the letter p. • Denote the probability of the complement of the event by the letter q. • p + q must equal 1 • q = 1 - p

23. Probability Related to Statistics • Probability makes statements about what will occur when samples are drawn from a known population. • Statistics describes how samples are to be obtained and how inferences are to be made about unknown populations.

24. Probability • The probability of an event is its long-run relative frequency. • While we may not be able to predict a particular individual outcome, we can talk about what will happen in the long run. • For any random phenomenon, each attempt, or trial, generates an outcome. • Something happens on each trial, and we call whatever happens the outcome. • These outcomes are individual possibilities, like the number we see on top when we roll a die.

25. Probability (cont.) • Sometimes we are interested in a combination of outcomes (e.g., a die is rolled and comes up even). • A combination of outcomes is called an event. • When thinking about what happens with combinations of outcomes, things are simplified if the individual trials are independent. • Roughly speaking, this means that the outcome of one trial doesn’t influence or change the outcome of another. • For example, coin flips are independent.

26. Independent Events The occurrence (or non-occurrence) of one event does not change the probability that the other event will occur.

27. If events A and B are independent, P(A and B) = P(A) • P(B)

28. Formal Probability (cont.) Multiplication Rule: • For two independent events A and B, the probability that both A and B occur is the product of the probabilities of the two events. • P(AandB) = P(A) x P(B), provided that A and B are independent.

29. Formal Probability (cont.) Multiplication Rule: • Two independent events A and B are not disjoint, provided the two events have probabilities greater than zero:

30. Conditional Probability • If events are dependent, the occurrence of one event changes the probability of the other. • The notation P(A|B) is read “the probability of A, given B.” • P(A, given B) equals the probability that event A occurs, assuming that B has already occurred.

31. For Dependent Events: • P(A and B) = P(A) • P(B, given A) • P(A and B) = P(B) • P(A, given B)

32. The Multiplication Rules: • For independent events: P(A and B) = P(A) • P(B) • For dependent events: P(A and B) = P(A) • P(B, given A) P(A and B) = P(B) • P(A, given B)

33. For independent events:P(A and B) = P(A) • P(B) When choosing two cards from two separate decks of cards, find the probability of getting two fives. P(two fives) = P(5 from first deck and 5 from second) =

34. For dependent events:P(A and B) = P(A) • P(B, given A) When choosing two cards from a deck without replacement, find the probability of getting two fives. P(two fives) = P(5 on first draw and 5 on second) =

35. “And” versus “or” • And means both events occur together. • Or means that at least one of the events occur.

36. For any events A and B, P(A or B) = P(A) + P(B) – P(A and B)

37. When choosing a card from an ordinary deck, the probability of getting a five or a red card: P(5 ) + P(red) – P(5 and red) =

38. When choosing a card from an ordinary deck, the probability of getting a five or a six: P(5 ) + P(6) – P(5 and 6) =

39. For any mutually exclusive events A and B, P(A or B) = P(A) + P(B)

40. When rolling an ordinary die: P(4 or 6) =

41. Survey results: P(male and college grad) = ?

42. Survey results: P(male and college grad) =

43. Survey results: P(male or college grad) = ?

44. Survey results: P(male or college grad) =

45. Survey results: P(male, given college grad) = ?

46. Survey results: P(male, given college grad) =

47. Can you write three (3) probability questions from the following data? The makers of the movie Titanic imply that lower-class passengers were treated unfairly when the lifeboats were being filled. We want to determine whether that portrayal is accurate. The following table contains the survival data by passenger class for the 1316 passengers. Now trade papers with someone and see if you can answer their questions correctly ; )

48. Counting Techniques • Tree Diagram • Multiplication Rule • Permutations • Combinations

49. Tree Diagram a method of listing outcomes of an experiment consisting of a series of activities

50. Tree diagram for the experiment of tossing two coins H H T start H T T