Probability addition rule
- 1. 1 MARIO F. TRIOLA Essentials of STATISTICS Section 3-3 Addition Rule
- 2. 2 Example: Let event A represent a woman and B represent blue eyes. A or B is a compound event representing the set of people who are women or who have blue eyes. A and B is a compound event that represents the set of people who are women AND have blue eyes (i.e. blue eyed women) A compound event is any event that combines two or more simple events.
- 3. 3 Venn Diagrams Women blue eyes Let the circle on the left represent the characteristic “subject is a woman” Let the circle on the right represent the characteristic “subject has blue eyes”
- 4. 4 Venn Diagrams– “or” Women blue eyes Together, the blue and yellow circles represent the compound event “is a woman or has blue eyes.”
- 5. 5 Venn Diagrams Women blue eyes The intersection of the two circles represents the event “subject is a woman and has blue eyes.”
- 6. 6 A={accountant} B = {is a downhill skier} A or B ? A and B? Interpret these Events:
- 7. 7 A={accountant} B = {is a downhill skier} A or B ? Is an accountant or is a downhill skier A and B? Is an accountant and a downhill skier Interpret these Events:
- 8. 8 When finding the probability that event A occurs or event B occurs, find 1. the total number of ways A can occur and 2. the number of ways B can occur, 3. but find the total in such a way that no outcome is counted more than once. General Addition Rule
- 9. 9 P(A or B) = P(A) + P(B) - P(A and B) where P(A and B) denotes the probability that A and B both occur at the same time. Formal Addition Rule If P(A) = .5 and P(B) = .2 and P(A and B) = .1, then, P(A or B) = 0.5 + 0.2 – 0.1 = 0.6
- 10. 10 Example: Addition Rule In a study of 82 young drivers, 39 were men who at some point in time were ticketed for traffic violations. 11 were men who were never ticketed. 8 were women who were ticketed, and 24 were women who were never ticketed. If one of these subjects is randomly selected, what is the probability of: (a)Getting a man or someone who was ticketed? (b)Getting a man or someone who was NOT ticketed? (c)Getting a woman or someone who was ticketed?
- 11. 11 Example: Addition Rule In a study of 82 young drivers, 39 were men who at some point in time were ticketed for traffic violations. 11 were men who were never ticketed. 8 were women who were ticketed, and 24 were women who were never ticketed. P(man) = 50 men / 82 surveyed = 0.6098 P(woman) = 32 women / 82 surveyed = 0.3902 P(ticketed) = 47 ticketed / 82 surveyed = 0.5732 P(never ticketed) = 35 never ticketed / 82 surveyed = 0.4268
- 12. 12 Example: Addition Rule If one of these subjects is randomly selected, what is the probability of: (a) Getting a man or someone who was ticketed? P(man or ticketed) = P(man) + P(ticketed) – P(man and ticketed) = 50/82 + 47/82 - 39/82 = 58/82 (b) Getting a man or someone who was NOT ticketed? P(man or not ticketed) = P(man) + P(not ticketed) – P(man and not ticketed) = 50/82 + 35/82 – 11/82 = 74/82 (c) Getting a woman or someone who was ticketed? P(woman or ticketed) = P(woman) + P(ticketed) – P(woman and ticketed) = 32/82 + 47/82 - 8/82 = 71/82
- 13. 13 Mutually Exclusive Events A and B are mutually exclusive if they cannot occur simultaneously.
- 14. 14 Mutually Exclusive Events A and B are mutually exclusive if they cannot occur simultaneously. P(A) P(B) P(A and B) NOT mutually exclusive!
- 15. 15 Definition Events A and B are mutually exclusive if they cannot occur simultaneously. P(A) P(B) Events A and B are Mutually exclusive Nothing in common
- 16. 16 Mutually Exclusive Events If two events A and B are mutually exclusive, they cannot occur at the same time. The implication is that it is impossible for these two events A and B to occur together, so P(A and B) = 0
- 17. 17 Mutually Exclusive Events If two events are mutually exclusive, this simplifies the addition rule. P(A or B) = P(A) + P(B) - P(A and B) = P(A) + P(B) - 0 = P(A) + P(B)
- 18. 18 Applying the Addition Rule P(A or B) Addition Rule Are A and B mutually exclusive ? P(A or B) = P(A)+ P(B) - P(A and B) P(A or B) = P(A) + P(B) Yes No
- 19. 19 Find the probability of randomly selecting a man or a boy. Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 56 2223 Contingency Table
- 20. 20 Find the probability of randomly selecting a man or a boy. Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 56 2223 Contingency Table
- 21. 21 Find the probability of randomly selecting a man or a boy. P(man or boy) = 1692 + 64 = 1756 = 0.790 2223 2223 2223 Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 56 2223 Contingency Table
- 22. 22 Find the probability of randomly selecting a man or a boy. P(man or boy) = 1692 + 64 = 1756 = 0.790 2223 2223 2223 Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 56 2223 Contingency Table * Mutually Exclusive *
- 23. 23 Find the probability of randomly selecting a man or someone who survived. Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 56 2223 Contingency Table
- 24. 24 Find the probability of randomly selecting a man or someone who survived. Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 56 2223 Contingency Table
- 25. 25 Find the probability of randomly selecting a man or someone who survived. P(man or survivor) = 1692 + 706 - 332 = 1756 2223 2223 2223 2223 Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 56 2223 Contingency Table = 0.929
- 26. 26 Find the probability of randomly selecting a man or someone who survived. P(man or survivor) = 1692 + 706 - 332 = 1756 2223 2223 2223 2223 Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 1517 Total 1692 422 64 56 2223 Contingency Table * NOT Mutually Exclusive * = 0.929
- 28. 28 Complementary Events P(A) and P(A) are mutually exclusive
- 29. 29 Complementary Events P(A) and P(A) are mutually exclusive All simple events are either in A or A.
- 30. 30 Complementary Events If two events are mutually exclusive, then they cannot occur at the same time. The probabilities of these two events will make 1. P(A) + P(A) = 1
- 31. 31 Rules of Complementary Events P(A) + P(A) = 1
- 32. 32 P(A) Rules of Complementary Events P(A) + P(A) = 1 = 1 - P(A)
- 33. 33 P(A) + P(A) = 1 = 1 - P(A) P(A) = 1 - P(A) P(A) Rules of Complementary Events
- 34. 34 Example: When a baby is born in the United States, the true probability that the child is a boy is 0.512. What is the true probability that the child is a girl? The events are complementary. One cannot be a boy and a girl at the same time. P(girl) = 1 – P(boy) = 1 – 0.512 = 0.488
- 35. 35 Example: A poll shows that 61% of Americans say they believe that life exists elsewhere in the galaxy. What is the probability that someone does not have that belief? The events are complementary. One cannot believe in life and not believe in life at the same time. P(not) = 1 – P(believes) = 1 – 0.61 = 0.39
- 36. 36 Example: Your alarm clock has a 97.5% probability of working on any given morning. What is the probability that it will NOT work? The events are complementary. The alarm cannot “work” and “not work” at the same time. P(not) = 1 – P(works) = 1 – 0.975 = 0.025
- 37. 37 Example: Suppose your use the principle of redundancy to wake you up in the morning. You set TWO alarms, both with 97.5% probability of working on any given morning. What is the probability that your system works? The events “works” and “fails” are two complementary events. P(works) = 1 – P(both fail) = 1 – (0.025)(0.025) = 0.999375 The redundant alarm system is almost certainly going to work every morning.
Editor's Notes
- #3: The last section dealt with simple events. This section deals with compound events. This text uses ‘or’ to mean the inclusive or, which mean either one or the other or both. The exclusive or is discussed only in exercise 27 of this section.
- #7: The last section dealt with simple events. This section deals with compound events. This text uses ‘or’ to mean the inclusive or, which mean either one or the other or both. The exclusive or is discussed only in exercise 27 of this section.
- #8: The last section dealt with simple events. This section deals with compound events. This text uses ‘or’ to mean the inclusive or, which mean either one or the other or both. The exclusive or is discussed only in exercise 27 of this section.
- #10: page 129 of text
- #14: Instructor should give some examples of some events which are mutually exclusive.
- #19: page 130 of text
- #20: Example page 130 of text.
- #30: Instructor should give examples of simple events A and not-A.
- #31: Discussion should be held about why P(A) + P(not A) equals 1.
- #32: page 132 of text
- #34: Equivalent forms of the Rule of Complementary Events The advantage of this rule will be illustrated in section 3-5.