1. Elementary Probability Theory
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.
Events with probability close to zero are less likely to occur.

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

4. Event A is certain to occur
If P(A) = 1
Event A is certain to occur

5. Event A is certain not to occur
If P(A) = 0
Event A is certain not to occur

6. Three methods to find probabilities:
Intuition
Relative frequency
Equally likely outcomes

7. Intuition Method of Determining Probability
Incorporates past experience, judgment, or opinion.
Is based upon level of confidence in the result
Example: “I am 95% sure that I will attend the party.”

8. Probability as Relative Frequency
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

9. 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:

10. Equally Likely Outcomes
No one result is expected to occur more frequently than any other.

11. 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. 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 actual) probability value.

14. Statistical Experiment or Statistical Observation
Any random activity that results in a definite outcome

15. Event
A collection of one or more outcomes of a statistical experiment or observation

16. Simple Event
An outcome of a statistical experiment that consists of one and only one of the outcomes of the experiment

17. Sample Space
The set of all possible distinct outcomes of an experiment
The sum of all probabilities of all simple events in a sample space must equal one.

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

19. For the experiment of rolling an ordinary die:
P(even number) =
P(result less than five) =
P(not getting a two) =

20. Complement of Event A the event that A does not occur
Notation for the complement of event A:

21. Event A and its complement Ac

22. Probability of a Complement
P(event A does not occur) =
P(Ac) = 1 – P(A)
So, P(A) + P(Ac) = 1

23. 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

24. 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.

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

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

27. Conditional Probability
If events are dependent, the fact that one occurs affects the probability of the other.
P(A, given B) equals the probability that event A occurs, assuming that B has already occurred.

28. General Multiplication Rule for Any Events:
P(A and B) = P(A) • P(B, given A)
P(A and B) = P(B) • P(A, given B)

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

30. 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) =

31. 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) =

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

33. The Event A and B

34. The Event A or B

35. General Addition Rule For any events A and B, P(A or B) =
P(A) + P(B) – P(A and B)

36. P(5 ) + P(red) – P(5 and red) =
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) =

37. 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) =

38. Mutually Exclusive Events
Events that are disjoint, cannot happen together.

39. Addition Rule for Mutually Exclusive Events
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. Counting Techniques Tree Diagram Multiplication Rule of Counting
Permutations
Combinations

48. Tree Diagram
a visual display of the total number of outcomes of an experiment consisting of a series of events

49. Tree diagram for selecting class schedules

50. Find the number of paths without constructing the tree diagram:
Experiment of rolling two dice, one after the other and observing any of the six possible outcomes each time .
Number of paths = 6 x 6 = 36

51. Multiplication Rule of Counting
For any series of events, if there are
n1 possible outcomes for event E1
and n2 possible outcomes for event E2, etc.
then there are
n1 X n2 X ··· X nm possible outcomes
for the series of events E1 through Em.

52. Area Code Example In the past, an area code consisted of 3 digits,
the first of which could be any digit from 2 through 9.
The second digit could be only a 0 or 1.
The last could be any digit.
How many different such area codes were possible?
8 · 2 · 10 = 160

53. Ordered Arrangements
In how many different ways could four items be arranged in order from first to last?
4 · 3 · 2 · 1 = 24

54. Factorial Notation n! is read "n factorial"
n! is applied only when n is a whole number.
n! is a product of n with each positive counting number less than n

55. Calculating Factorials
120
5! = 5 • 4 • 3 • 2 • 1 =
3! = 3 • 2 • 1 = 6

56. Definitions
1! = 1
0! = 1

57. Complete the Factorials:
4! =
10! =
6! =
15! = 24
3,628,800
720
x 1012

58. Permutations
A Permutation is an arrangement in a particular order of a group of items.
There are to be no repetitions of items within a permutation.

59. Listing Permutations
How many different permutations of the letters a, b, c are possible?
Solution: There are six different permutations:
abc, acb, bac, bca, cab, cba.

60. Listing Permutations
How many different two-letter permutations of the letters a, b, c, d are possible?
Solution: There are twelve different permutations:
ab, ac, ad, ba, ca, da, bc, bd, cb, db, cd, dc.

61. Permutation Formula
The number of ways to arrange in order n distinct objects, taking them r at a time, is:
where n and r are whole numbers and n > r.

62. Another Notation for Permutations

63. Find P7, 3

64. Applying the Permutation Formula6 12 30
336
210

65. Application of Permutations
A teacher has chosen eight possible questions for an upcoming quiz. In how many different ways can five of these questions be chosen and arranged in order from #1 to #5?
Solution: P8,5 =
= 8• 7 • 6 • 5 • 4 = 6720

66. A combination is a grouping
Combinations
A combination is a grouping
in no particular order
of items.

67. The number of combinations of n objects taken r at a time is:
Combination Formula
The number of combinations of n objects taken r at a time is:
where n and r are whole numbers and n > r.

68. Other Notations for Combinations:

69. Find C9, 3

70. Applying the Combination Formula10 35
C5, 3 = C7, 3 =
C3, 3 = C15, 2 =
C6, 2 = 1
105 15

71. Application of Combinations
A teacher has chosen eight possible questions for an upcoming quiz. In how many different ways can five of these questions be chosen if order makes no difference?
Solution: C8,5 =
= 56

72. Determining the Number of Outcomes of an Experiment
If the experiment consists of a series of stage with various outcomes, use the multiplication rule or a tree diagram.

73. Determining the Number of Outcomes of an Experiment
If the outcomes consist of ordered subgroups of r outcomes taken from a group of n outcomes use the permutation rule.

74. Determining the Number of Outcomes of an Experiment
If the outcomes consist of non-ordered subgroups of r items taken from a group of n items use the combination rule.