1. 7 Probability Experiments, Sample Spaces, and Events
Definition of Probability
Rules of Probability
Use of Counting Techniques in Probability
Conditional Probability and Independent Events
Bayes’ Theorem

2. Experiments, Sample Spaces, and Events
7.1
Experiments, Sample Spaces, and Events

3. Terminology Experiment
An experiment is an activity with observable results.
The results of an experiment are called outcomes of the experiment.

4. Examples Tossing a coin and observing whether it falls heads or tails
Rolling a die and observing which of the numbers 1, 2, 3, 4, 5, or 6 shows up
Testing a spark plug from a batch of 100 spark plugs and observing whether or not it is defective

5. Terminology Sample Point, Sample Space, and Event Sample point:
An outcome of an experiment
Sample space:
The set consisting of all possible sample points of an experiment
Event:
A subset of a sample space of an experiment

6. Example
Describe the sample space associated with the experiment of tossing a coin and observing whether it falls heads or tails.
What are the events of this experiment?
Solution
The two outcomes are heads and tails, and the required sample space is given by
S = {H, T}
where H denotes the outcome heads and T denotes the outcome tails.
The events of the experiment, the subsets of S, are
Ø, {H}, {T}, S
Example 1, page 354

7. Terminology Union of Two Events
The union of two events E and F is the event E  F.
Thus, the event E  F contains the set of outcomes of E and/or F.

8. Terminology Intersection of Two Events
The intersection of two events E and F is the event E  F.
Thus, the event E  F contains the set of outcomes common to E and F.

9. Terminology Complement of an Event
The complement of an event E is the event E c.
Thus, the event E c is the set containing all the outcomes in the sample space S that are not in E.

10. Example
Consider the experiment of rolling a die and observing the number that falls uppermost.
Let
S = {1, 2, 3, 4, 5, 6}
denote the sample space of the experiment and
E = {2, 4, 6} and F = {1, 3}
be events of this experiment.
Compute E  F. Interpret your results.
Solution
E  F = {1, 2, 3, 4, 6} and is the event that the outcome of the experiment is a 1, a 2, a 3, a 4, or a 6.
Example 2, page 355

11. Example
Consider the experiment of rolling a die and observing the number that falls uppermost.
Let
S = {1, 2, 3, 4, 5, 6}
denote the sample space of the experiment and
E = {2, 4, 6} and F = {1, 3}
be events of this experiment.
Compute E  F. Interpret your results.
Solution
E  F = Ø since there are no elements in common between the two sets E and F.
Example 2, page 355

12. Example
Consider the experiment of rolling a die and observing the number that falls uppermost.
Let
S = {1, 2, 3, 4, 5, 6}
denote the sample space of the experiment and
E = {2, 4, 6} and F = {1, 3}
be events of this experiment.
Compute F c. Interpret your results.
Solution
F c = {2, 4, 5, 6} is precisely the event that the event F does not occur.
Example 2, page 355

13. Terminology Mutually Exclusive Events
E and F are mutually exclusive if
E  F = Ø

14. Example
An experiment consists of tossing a coin three times and observing the resulting sequence of heads and tails.
Describe the sample space S of the experiment.
Determine the event E that exactly two heads appear.
Determine the event F that at least one head appears.
Example 3, page 356

15. S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Example
Solution
As the tree diagram demonstrates, the sample space is
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Example 2, page 355

16. Example Solution Scanning the sample space obtained
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
we see that the outcomes in which exactly two heads appear are given by the event
E = {HHT, HTH, THH}
We can also see that the outcomes in which at least one head appears are given by the event
F = {HHH, HHT, HTH, HTT, THH, THT, TTH}
Example 2, page 355

17. Applied Example: Movie Attendance
The manager of a local cinema records the number of patrons attending a first-run movie screening.
The theatre has a seating capacity of 500.
Determine an appropriate sample space for this experiment.
Describe the event E that fewer than 50 people attend the screening.
Describe the event F that the theatre is more than half full at the screening.
Solution
The number of patrons at the screening could run from to 500. Therefore, a sample space for this experiment is
S = {0, 1, 2, 3, …, 500}
Applied Example 5, page 357

18. Applied Example: Movie Attendance
The manager of a local cinema records the number of patrons attending a first-run movie screening.
The theatre has a seating capacity of 500.
Determine an appropriate sample space for this experiment.
Describe the event E that fewer than 50 people attend the screening.
Describe the event F that the theatre is more than half full at the screening.
Solution
The event E that fewer than 50 people attend the screening can be described as
E = {0, 1, 2, 3, …, 49}
Applied Example 5, page 357

19. Applied Example: Movie Attendance
The manager of a local cinema records the number of patrons attending a first-run movie screening.
The theatre has a seating capacity of 500.
Determine an appropriate sample space for this experiment.
Describe the event E that fewer than 50 people attend the screening.
Describe the event F that the theatre is more than half full at the screening.
Solution
The event F that the theatre is more than half full at the screening can be described as
F = {251, 252, …, 500}
Applied Example 5, page 357

20. Definition of Probability
7.2
Definition of Probability

21. Probability of an Event in a Uniform Sample Space
S = {s1, s2, … , sn}
is the sample space for an experiment in which the outcomes are equally likely, then we assign the probabilities
to each of the outcomes s1, s2, … , sn.

22. Example
A fair die is rolled, and the number that falls uppermost is observed.
Determine the probability distribution for the experiment.
Solution
The sample space for the experiment is
S = {1, 2, 3, 4, 5, 6}
and the simple events are accordingly given by the sets
{1}, {2}, {3}, {4}, {5}, and {6}
Since the die is assumed to be fair, the six outcomes are equally likely.
We therefore assign a probability of 1/6 to each of the simple events.
Example 1, page 364

23. Example
A fair die is thrown, and the number that falls uppermost is observed.
Determine the probability distribution for the experiment.
Solution
Thus, the probability distribution of these simple events is:
Simple
event
Probability
{1}
1/6
{2}
{3}
{4}
{5}
{6}
Example 1, page 364

24. Finding the Probability of Event E
Determine a sample space S associated with the experiment.
Assign probabilities to the simple events of S. If
E = {s1, s2, … , sn}
where {s1}, {s2}, {s3}, … , {sn} are simple events, then
P(E) = P(s1) + P(s2) + P(s3) + ··· + P(sn)
If E is the empty set, Ø, then P(E) = 0.

25. Applied Example: Rolling Dice
A pair of fair dice is rolled.
Calculate the probability that the two dice show the same number.
Calculate the probability that the sum of the numbers of the two dice is 6.
Applied Example 3, page 365

26. Applied Example: Rolling Dice
Solution
The sample space S of the experiment has 36 outcomes
S = {(1, 1), (1, 2), … , (6, 5), (6, 6)}
Both dice are fair, making each of the 36 outcomes equally likely, so we assign the probability of 1/36 to each simple event.
The event that the two dice show the same number is
E = {(1, 1), (2, 2) , (3, 3), (4, 4), (5, 5), (6, 6)}
Therefore, the probability that the two dice show the same number is given by
Six terms
Applied Example 3, page 365

27. Applied Example: Rolling Dice
Solution
The event that the sum of the numbers of the two dice is 6 is given by
E6 = {(1, 5), (2, 4) , (3, 3), (4, 2), (5, 1)}
Therefore, the probability that the sum of the numbers on the two dice is 6 is given by
Applied Example 3, page 365

28. 7.3 Rules of Probability Property 1. P(E)  0 for every E.
Property 2. P(S) = 1.
Property 3. If E and F are mutually exclusive
(E  F = Ø), then
P(E  F) = P(E) + P(F)
Property 4. If E and F are any two events of an
experiment, then
P(E  F) = P(E) + P(F) – P(E  F)
Property 5. If E is an event of an experiment and
Ec denotes the complement of E, then
P(Ec) = 1 – P(E)

29. Properties of the Probability Function
Property 1. P(E)  0 for every E.
Property 2. P(S) = 1.
Property 3. If E and F are mutually exclusive (E  F = Ø), then
P(E  F) = P(E) + P(F)

30. Applied Example: SAT Verbal Scores
The superintendent of a metropolitan school district has estimated the probabilities associated with the SAT verbal scores of students from that district.
The results are shown in the table below.
If a student is selected at random, find the probability that his or her SAT verbal score will be
More than 400.
Less than or equal to 500.
Greater than 400 but less than or equal to 600.
Score, x
Probability
x > 700
.01
600 < x  700
.07
500 < x  600
.19
400 < x  500
.23
300 < x  400
.31
x  300
Applied Example 1, page 372

31. Applied Example: SAT Verbal Scores
Solution
Let A, B, C, D, E, and F denote, respectively, the mutually exclusive events listed in the table below.
The probability that the student’s score will be more than 400 is given by
P(D  C  B  A) = P(D) + P(C) + P(B) + P(A)
= = .50
Score, x
Probability
x > 700
.01
600 < x  700
.07
500 < x  600
.19
400 < x  500
.23
300 < x  400
.31
x  300 A B C D E F
Applied Example 1, page 372

32. Applied Example: SAT Verbal Scores
Solution
Let A, B, C, D, E, and F denote, respectively, the mutually exclusive events listed in the table below.
The probability that the student’s score will be less than or equal to 500 is given by
P(D  E  F) = P(D) + P(E) + P(F)
= = .73
Score, x
Probability
x > 700
.01
600 < x  700
.07
500 < x  600
.19
400 < x  500
.23
300 < x  400
.31
x  300 A B C D E F
Applied Example 1, page 372

33. Applied Example: SAT Verbal Scores
Solution
Let A, B, C, D, E, and F denote, respectively, the mutually exclusive events listed in the table below.
The probability that the student’s score will be greater than 400 but less than or equal to 600 is given by
P(C  D) = P(C) + P(D)
= = .42
Score, x
Probability
x > 700
.01
600 < x  700
.07
500 < x  600
.19
400 < x  500
.23
300 < x  400
.31
x  300 A B C D E F
Applied Example 1, page 372

34. Properties of the Probability Function
Addition Rule
Property 4. If E and F are any two events of an experiment, then
P(E  F) = P(E) + P(F) – P(E  F)

35. Example A card is drawn from a shuffled deck of 52 playing cards.
What is the probability that it is an ace or a spade?
Solution
Let E denote the event that the card drawn is an ace, and let F denote the event that the card drawn is a spade.
Then,
Note that E and F are not mutually exclusive events:
E  F is the event that the card drawn is an ace of spades.
Consequently,
Example 2, page 373

36. Example A card is drawn from a shuffled deck of 52 playing cards.
What is the probability that it is an ace or a spade?
Solution
The event that a card drawn is an ace or a spade is E  F, with probability given by
Example 2, page 373

37. Applied Example: Quality Control
The quality-control department of Vista Vision, manufacturer of the Pulsar plasma TV, has determined from records obtained from the company’s service centers that 3% of the sets sold experience video problems, 1% experience audio problems, and 0.1% experience both video and audio problems before the expiration of the warranty.
Find the probability that a plasma TV purchased by a consumer will experience video or audio problems before the warranty expires.
Applied Example 3, page 373

38. Applied Example: Quality Control
Solution
Let E denote the event that a plasma TV purchased will experience video problems within the warranty period, and let F denote the event that a plasma TV purchased will experience audio problems within the warranty period.
Then,
P(E) = P(F) = P(E  F) = .001
The event that a plasma TV purchased will experience video or audio problems before the warranty expires is
E  F, and the probability of this event is given by
Applied Example 3, page 373

39. Properties of the Probability Function
Rule of Complements
Property 5. If E is an event of an experiment and Ec denotes the complement of E, then
P(Ec) = 1 – P(E)

40. Applied Example: Warranties
What is the probability that a Pulsar plasma TV (from the last example) bought by a consumer will not experience video or audio problems before the warranty expires?
Solution
Let E denote the event that a plasma TV bought by a consumer will experience video or audio problems before the warranty expires.
Then, the event that the plasma TV will not experience either problem before the warranty expires is given by Ec, with probability
Applied Example 4, page 375

41. Use of Counting Techniques in Probability
7.4
Use of Counting Techniques in Probability

42. Computing the Probability of an Event in a Uniform Sample Space
Let S be a uniform sample space and let E be any event. Then,

43. Example An unbiased coin is tossed six times.
Find the probability that the coin will land heads
Exactly three times.
At most three times.
On the first and the last toss.
Example 1, page 381

44. Example
Solution
Each outcome of the experiment may be represented as a sequence of heads and tails.
Using the generalized multiplication principle, we see that the number of outcomes of this experiment is 26, or 64.
Let E denote the event that the coin lands heads exactly three times.
Since there are C(6, 3) ways this can occur, we see that the required probability is
Example 1, page 381

45. Example
Solution
Let F denote the event that the coin lands heads at most three times.
Then, n(F) is given by the sum of the number of ways the coin lands heads zero times (no heads), exactly once, exactly twice, and exactly three times. That is,
Thus, the required probability is
Example 1, page 381

46. Example
Solution
Let H denote the event that the coin lands heads on the first and the last toss.
Then,
Therefore, the required probability is
Example 1, page 381

47. Conditional Probability and Independent Events
7.5
Conditional Probability and Independent Events

48. Conditional Probability of an Event
If A and B are events in an experiment and
P(A)  0, then the conditional probability that the event B will occur given that the event A has already occurred is

49. Example
A pair of fair dice is rolled. What is the probability that the sum of the numbers falling uppermost is 7 if it is known that exactly one of the numbers is a 5?
Solution
Let A denote the event that exactly one of the numbers is a 5 and let B denote the event that the sum of the numbers falling uppermost is 7. Thus, so
Example 2, page 390

50. Example
A pair of fair dice is rolled. What is the probability that the sum of the numbers falling uppermost is 7 if it is known that exactly one of the numbers is a 5?
Solution
Since the dice are fair, each outcome of the experiment is equally likely; therefore,
Thus, the probability that the sum of the numbers falling uppermost is 7 given that exactly one of the numbers is a 5 is given by
Example 2, page 390

51. Applied Example: Color Blindness
In a test conducted by the U.S. Army, it was found that of 1000 new recruits (600 men and 400 women), 50 of the men and 4 of the women were red-green color-blind.
Given that a recruit selected at random from this group is red-green color-blind, what is the probability that the recruit is a male?
Solution
Let C denote the event that a randomly selected subject is red-green color-blind, and let M denote the event that the subject is a male recruit.
Applied Example 3, page 390

52. Applied Example: Color Blindness
In a test conducted by the U.S. Army, it was found that of 1000 new recruits (600 men and 400 women), 50 of the men and 4 of the women were red-green color-blind.
Given that a recruit selected at random from this group is red-green color-blind, what is the probability that the recruit is a male?
Solution
Since 54 out of 1000 subjects are color-blind, we see that
Furthermore, since 50 of the subjects are colorblind and male, we see that
Applied Example 3, page 390

53. Applied Example: Color Blindness
In a test conducted by the U.S. Army, it was found that of 1000 new recruits (600 men and 400 women), 50 of the men and 4 of the women were red-green color-blind.
Given that a recruit selected at random from this group is red-green color-blind, what is the probability that the recruit is a male?
Solution
Therefore, the probability that a subject is male given that the subject is red-green color-blind is given by
Applied Example 3, page 390

54. Product Rule

55. Example
Two cards are drawn without replacement from a well-shuffled deck of 52 playing cards.
What is the probability that the first card drawn is an ace and the second card drawn is a face card?
Solution
Let A denote the event that the first card drawn is an ace, and let F denote the event that the second card drawn is a face card.
Then,
Example 5, page 392

56. Example
Two cards are drawn without replacement from a well-shuffled deck of 52 playing cards.
What is the probability that the first card drawn is an ace and the second card drawn is a face card?
Solution
After drawing the first card, there are 51 cards left in the deck, of which 12 are face cards.
Therefore, the probability of drawing a face card given that the first card drawn was an ace is given by
Example 5, page 392

57. Example
Two cards are drawn without replacement from a well-shuffled deck of 52 playing cards.
What is the probability that the first card drawn is an ace and the second card drawn is a face card?
Solution
By the product rule, the probability that the first card drawn is an ace and the second card drawn is a face card is given by
Example 5, page 392

58. Independent Events
If A and B are independent events, then

59. Test for the Independence of Two Events
Two events A and B are independent if and only if

60. Example
Consider the experiment consisting of tossing a fair coin twice and observing the outcomes.
Show that obtaining heads on the first toss and tails on the second toss are independent events.
Solution
Let A denote the event that the outcome of the first toss is a head, and let B denote the event that the outcome of the second toss is a tail.
The sample space of the experiment is so
Example 8, page 396

61. Example
Consider the experiment consisting of tossing a fair coin twice and observing the outcomes.
Show that obtaining heads in the first toss and tails in the second toss are independent events.
Solution
Next, we compute
and observe that the test for independent events is satisfied:
Example 8, page 396

62. Applied Example: Medical Survey
A survey conducted by an independent agency for the National Lung Society found that, of 2000 women, 680 were heavy smokers and 50 had emphysema.
Of those who had emphysema, 42 were heavy smokers.
Using the data in this survey, determine whether the events being a heavy smoker and having emphysema are independent events.
Solution
Let A denote the event that a woman is a heavy smoker, and let B denote the event that a woman has emphysema.
Then, the probabilities that a woman is a heavy smoker, has emphysema, or both are given by, respectively,
Applied Example 9, page 396

63. Applied Example: Medical Survey
A survey conducted by an independent agency for the National Lung Society found that, of 2000 women, 680 were heavy smokers and 50 had emphysema.
Of those who had emphysema, 42 were heavy smokers.
Using the data in this survey, determine whether the events being a heavy smoker and having emphysema are independent events.
Solution
Next, we see that the test for independent events is not satisfied: so
and conclude that A and B are not independent events.
Applied Example 9, page 396

64. 7.6
Bayes’ Theorem

65. P(E)  0 and P(Ai)  0 for 1 ≤ i ≤ n
Bayes’ Theorem
Let A1, A2, … , An be a partition of a sample space S, and let E be an event of the experiment such that
P(E)  0 and P(Ai)  0 for 1 ≤ i ≤ n
Then the a posteriori probability
is given by

66. Applied Example: Quality Control
The panels for the Pulsar widescreen LCD HDTVs are manufactured in three locations and then shipped to the main plant of Vista Vision for final assembly.
Plants A, B, and C supply 50%, 30%, and 20%, respectively, of the panels used by Vista Vision.
The quality-control department of the company has determined that 1% of the panels produced by plant A are defective, whereas 2% of the panels produced by plants B and C are defective.
If a Pulsar widescreen TV is selected at random and the panel is found to be defective, what is the probability that the panel was manufactured in plant C?
Applied Example 1, page 404

67. Applied Example: Quality Control
Solution
Let A, B, and C denote the event that the set chosen has a panel manufactured in plant A, plant B, or plant C, respectively.
Also, let D denote the event that a set has a defective panel.
We can draw a tree diagram with this information:
Applied Example 1, page 404

68. Applied Example: Quality Control
Solution
Next, using Bayes’ theorem, we find that the required a posteriori probability is given by
Applied Example 1, page 404

69. End of Chapter