Download C.2 Probability Computations

APPENDIX C.2
C.2
Probability Computations
A7
Probability Computations
■ Find the probability of mutually exclusive events.
■ Find the probability of the complement of an event.
■ Find the odds of an event.
Mutually Exclusive Events
In Appendix C.1, you looked at a formula for calculating simple probabilities. In
this section, you will study formulas for calculating probabilities of more
involved events and combinations of events. First, you will study a formula for
the probability of the union of two disjoint (or mutually exclusive) events.
Two events A and B (from the same sample space) are mutually exclusive if
A and B have no outcomes in common. That is, A 傽 B ⫽ ⭋. For instance, if two
dice are tossed, the event A of rolling a total of 6 and the event B of rolling a total
of 9 are mutually exclusive because the following sets have no elements in common. (The ordered pairs represent the numbers on the dice.)
A ⫽ 再共1, 5兲, 共2, 4兲, 共3, 3兲, 共4, 2兲, 共5, 1兲冎
B ⫽ 再共3, 6兲, 共4, 5兲, 共5, 4兲, 共6, 3兲冎
To find the probability that one or the other of two mutually exclusive events
will occur, you can add their individual probabilities, as indicated in Figure C.4.
B
A
P(A ∪ B) = P(A) + P(B)
FIGURE C.4
Probability of Mutually Exclusive Events
If A and B are mutually exclusive events, then for a given experiment the
probability that A or B will occur is
P共A 傼 B兲 ⫽ P共A兲 ⫹ P共B兲.
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APPENDIX C
Probability and Probability Distributions
Example 1
Finding the Probability of Mutually Exclusive
Events
A card is drawn from a standard deck of 52 cards. What is the probability that it
is a face card (event A) or a 4 (event B)?
The two events are mutually exclusive because no one card can be
both a face card and a 4. Because
SOLUTION
P共A兲 ⫽
n共A兲 12
3
⫽
⫽
n共S兲
52 13
P共B兲 ⫽
n共B兲
4
1
⫽
⫽
n共S兲
52 13
and
it follows that
P共A 傼 B兲 ⫽ P共A兲 ⫹ P共B兲
3
1
4
⫽
⫹
⫽ .
13 13 13
So, you can conclude that the probability of selecting a face card or a 4 is
4
13 ⬇ 0.308.
✓CHECKPOINT 1
A card is drawn from a standard deck of 52 cards. What is the probability that
the card is a jack or an ace? ■
Three events A, B, and C in the same sample space are mutually exclusive if
A 傽 B ⫽ ⭋,
A 傽 C ⫽ ⭋,
B 傽 C ⫽ ⭋.
The probability that A, B, or C will occur is given by
P共A 傼 B 傼 C兲 ⫽ P共A兲 ⫹ P共B兲 ⫹ P共C兲.
This pattern can be applied to four or more events that are mutually exclusive. That is, if A1, A2, A3, . . . , An are mutually exclusive events in the same
sample space then
P共A1 傼 A 2 傼 A3 傼 . . . 傼 An兲 ⫽ P共A1兲 ⫹ P共A 2兲 ⫹ P共A3兲 ⫹ . . . ⫹ P共An兲.
APPENDIX C.2
Example 2
Probability Computations
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Finding the Probability of Mutually Exclusive
Events
The weights of adult Lhasa Apsos at a breeding facility are shown in the table.
Weight (in pounds)
6–8
9–11
12–14
15–17
18–20
21–23
Number of dogs
0
2
7
5
3
1
What is the probability that an adult Lhasa Apso chosen at random will weigh
between 12 and 20 pounds?
SOLUTION
Let A, B, and C represent the following events.
A ⫽ weight between 12 and 14 pounds
B ⫽ weight between 15 and 17 pounds
C ⫽ weight between 18 and 20 pounds
Because these three events are mutually exclusive, the probability that the adult
Lhasa Apso’s weight will fall between 12 and 20 pounds is
P 共A 傼 B 傼 C兲 ⫽ P共A兲 ⫹ P共B兲 ⫹ P共C兲
7
5
3
⫽
⫹
⫹
18 18 18
15
⫽
18
⬇ 0.833
✓CHECKPOINT 2
In Example 2, find the probability that an adult Lhasa Apso chosen at random
weighs between 9 and 17 pounds. ■
If a sample space S is subdivided into a collection of mutually exclusive
events A1, A2 , A3 , . . . , An such that
A1 傼 A 2 傼 A3 傼 . . . 傼 An ⫽ S
then the sum of the probabilities of the events must be 1. This Addition Principle
is summarized as follows.
Addition Principle
If A1, A 2, A3 , . . . , An are mutually exclusive events in a sample space S
such that
A1 傼 A 2 傼 A 3 傼 . . . 傼 An ⫽ S,
then
P共A1兲 ⫹ P共A2兲 ⫹ P共A3兲 ⫹ . . . ⫹ P共A n兲 ⫽ 1.
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APPENDIX C
Probability and Probability Distributions
Example 3
Applying the Addition Principle
Two six-sided dice are tossed. Construct a table showing the probabilities for the
various totals of the two dice. Then show that the sum of these probabilities is 1.
SOLUTION The outcomes representing various totals are shown in the following
table, which is called a frequency distribution.
Number of
Outcomes
Total
Outcomes
2
3
4
5
6
7
8
9
10
11
12
共1, 1兲
共1, 2兲, 共2, 1兲
共1, 3兲, 共2, 2兲, 共3, 1兲
共1, 4兲, 共2, 3兲, 共3, 2兲, 共4, 1兲
共1, 5兲, 共2, 4兲, 共3, 3兲, 共4, 2兲, 共5, 1兲
共1, 6兲, 共2, 5兲, 共3, 4兲, 共4, 3兲, 共5, 2兲, 共6, 1兲
共2, 6兲, 共3, 5兲, 共4, 4兲, 共5, 3兲, 共6, 2兲
共3, 6兲, 共4, 5兲, 共5, 4兲, 共6, 3兲
共4, 6兲, 共5, 5兲, 共6, 4兲
共5, 6兲, 共6, 5兲
共6, 6兲
1
2
3
4
5
6
5
4
3
2
1
From this frequency distribution, you can construct the following table showing
the probability of each of the various totals. This table is called a probability
distribution.
✓CHECKPOINT 3
Five coins are tossed. Construct a
table showing the probabilities for
the number of heads that turn up.
Then show that the sum of the
probabilities is 1. ■
A∩B
A
B
A
Total
2
3
4
5
6
7
8
9
10 11 12
Probability
1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
FIGURE C.5
1
36
Note that the events in this probability distribution are mutually exclusive and
represent every possible total (or outcome). From the Addition Principle, you can
see that the sum of the given probabilities is 1.
1
2
3
4
5
6
5
4
3
2
1
⫹
⫹
⫹
⫹
⫹
⫹
⫹
⫹
⫹
⫹
⫽1
36 36 36 36 36 36 36 36 36 36 36
Be sure you see that the probability of A 傼 B is equal to the sum of the probabilities of A and B only if A and B are mutually exclusive. For instance, the
probability of drawing a heart (event A) or a face card (event B) from a standard
deck of 52 cards is not equal to P共A兲 ⫹ P共B兲 because A and B are not mutually
exclusive.
To find the probability of A 傼 B, where A and B are not necessarily mutually
exclusive events, use the fact that
n共A 傼 B兲 ⫽ n共A兲 ⫹ n共B兲 ⫺ n共A 傽 B兲.
From this, it follows that
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
2
36
P共A 傼 B兲 ⫽ P共A) ⫹ P共B兲 ⫺ P共A 傽 B兲
as summarized in the following box. (See Figure C.5)
APPENDIX C.2
Probability Computations
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Probability of the Union of Two Events
If A and B are events in the same sample space S, then the probability of
A 傼 B is
P共A 傼 B兲 ⫽ P共A兲 ⫹ P共B兲 ⫺ P共A 傽 B兲.
Example 4
Finding the Probability of the Union of
Two Events
One card is selected from a standard deck of 52 playing cards. What is the
probability that the card is either a heart or a face card.
SOLUTION
Hearts
A♥ 2♥
n(A ∩ B) = 3
3♥
5♥
4♥
6♥ 7♥
J♥ J♠
J♦
Q♥
8♥ 9♥
Q♠
K♥
J♣
10♥
Q♦
K♠ K♦
Q♣
Face cards
K♣
FIGURE C.6
Because the deck has 13 hearts, the probability of selecting a heart
(event A) is
P共A兲 ⫽
13 1
⫽ .
52 4
Because the deck has 12 face cards, the probability of selecting a face card (event
B) is
P共B兲 ⫽
12
3
⫽ .
52 13
One card is selected from a standard deck of 52 playing cards.
What is the probability that the
card is either a king or a face
card? ■
Selecting a face card
Because three of the cards are hearts and face cards (See Figure C.6) it follows
that
P共A 傽 B兲 ⫽
✓CHECKPOINT 4
Selecting a heart
3
.
52
Finally, applying the formula for the probability of the union of two events, you
can conclude that the probability of selecting a heart or a face card is
P共A 傼 B兲 ⫽ P共A兲 ⫹ P共〉兲 ⫺ P共A 傽 B兲
⫽
13 12
3
22
⫹
⫺
⫽
⬇ 0.423
52 52 52 52
The Complement of an Event
The complement of an event A is the collection of all outcomes in the sample
space that are not in A. The complement of event A is denoted by A⬘. Because
P共A 傼 A⬘ 兲 ⫽ 1 and because A and A⬘ are mutually exclusive, it follows that
P共A兲 ⫹ P共A⬘ 兲 ⫽ 1. So, the probability of A⬘ is P共A⬘ 兲 ⫽ 1 ⫺ P共A兲.
Probability of the Complement of an Event
If A is an event, then the probability of the complement of A is given by
P共A⬘ 兲 ⫽ 1 ⫺ P共A兲.
A12
APPENDIX C
Probability and Probability Distributions
For instance, if the probability of winning a game is
1
1 3
P共A兲 ⫽ , then the probability of losing the game is P共A⬘ 兲 ⫽ 1 ⫺ ⫽ .
4
4 4
Example 5
Finding the Probability of the Complement of
an Event
The age distribution for the population of the United States (in 2005) is shown in
the table. (Source: U.S. Census Bureau)
Age Bracket
Number of People
Age Bracket
Number of People
Under 5 years
20,304,000
40–44 years
22,861,000
5–13 years
36,087,000
45–49 years
22,485,000
14–17 years
17,079,000
50–54 years
19,998,000
18–24 years
29,307,000
55–64 years
30,356,000
25–29 years
20,066,000
65–74 years
18,640,000
30–34 years
20,077,000
18,150,000
35–39 years
21,002,000
75 years and
older
Total
296,412,000
If a person is chosen at random from this population, what is the probability that
the person is (a) under 18? (b) 18 or older?
SOLUTION
a. The number of people who are under 18 (event A) is
n共A兲 ⫽ 20,304,000 ⫹ 36,087,000 ⫹ 17,079,000 ⫽ 73,470,000
So, the probability of choosing a person who is under 18 is
P共A兲 ⫽
✓CHECKPOINT 5
In Example 5, what is the probability that the person is (a) under 25?
(b) 25 or older? ■
73,470,000
⬇ 0.248.
296,412,000
b. Because the event corresponding to choosing a person who is 18 or older is the
complement of A, it follows that the probability of choosing a person who is
18 or older is
P共A⬘ 兲 ⫽ 1 ⫺ P共A兲 ⬇ 1 ⫺ 0.248 ⫽ 0.752.
In Example 5, note the convenience of using the formula for the complement
of an event. Without this formula, you would have to determine the number of
outcomes in event A⬘ by adding many terms.
If A and B are any two events in the same sample space, then it must be true
that
共A 傽 B兲 傽 共A⬘ 傽 B兲 ⫽ ⭋.
APPENDIX C.2
Probability Computations
A13
So 共A 傽 B兲 and 共A⬘ 傽 B兲 are mutually exclusive. Moreover, because
A
B
共A 傽 B兲 傼 共A⬘ 傽 B兲 ⫽ B
it follows that
P共A 傽 B兲 ⫹ P共A⬘ 傽 B兲 ⫽ P共B兲.
A
A∩B
A′ ∩ B
P(A ∩ B) + P(A′ ∩ B) = P(B)
FIGURE C.7
(See Figure C.7.)
Example 6
Finding the Probability of (A⬘ 傽 B)
A college has an undergraduate enrollment of 3000 students. Of these, 860 are
business majors (event A) and 1400 are women (event B). Of the business majors,
375 are women. The college newspaper staff is conducting a poll and selects
undergraduate students at random to answer a survey.
a. What is the probability a student who is selected to participate in the poll is a
woman and a business major?
b. What is the probability a student who is selected to participate in the poll is a
woman and not a business major?
SOLUTION
a. The probability that a student who is selected is a woman and a business
major is
✓CHECKPOINT 6
In Example 6, what is the
probability a student is selected to
participate in the poll is (a) a man
and a business major? (b) a man
and not a business major? ■
P共A 傽 B兲 ⫽
375
1
⫽ .
3000 8
b. Because the probability that a student who is selected is a woman is
P共B兲 ⫽ 1400兾3000 ⫽ 7兾15, the probability that the student is a woman and
not a business major is
P共A⬘ 傽 B兲 ⫽ P共B兲 ⫺ P共A 傽 B兲 ⫽
7
1
41
⫺ ⫽
⬇ 0.342.
15 8 120
Odds
Probabilities are sometimes given in terms of odds, which is the ratio of P共A兲
to P共A⬘ 兲. For instance, the odds that a horse will win a race might be stated as
2 to 1.
Odds
If A is an event, then the odds that A will occur are given by
P共A兲
,
P共A⬘ 兲
P共A⬘ 兲 ⫽ 0.
Odds are usually read as “P共A兲 to P共A⬘ 兲.”
A14
APPENDIX C
Probability and Probability Distributions
Example 7
Finding the Odds That an Event Will Occur
Two six-sided dice are tossed. What are the odds that the total of the two dice will
be 7?
SOLUTION From Example 3, you know that the probability that the total will be
7 (event A) is
P共A兲 ⫽
6
1
⫽ .
36 6
So, the probability that the total will be a number other than 7 is
P共A⬘ 兲 ⫽ 1 ⫺
1 5
⫽ .
6 6
The odds that the total will be 7 are given by
odds ⫽
P共A兲
1兾6 1
⫽
⫽
P共A⬘ 兲 5兾6 5
which means that the odds are 1 to 5.
✓CHECKPOINT 7
Two six-sided dice are tossed. What are the odds that the total of the two dice
will be 8? ■
If you are given the odds that an event A will occur, then you can determine
the probability that A will occur by the following formula.
Finding Probability from the Odds
If the odds that an event A will occur are n to m, then the probability that A
will occur is
P共A兲 ⫽
n
.
m⫹n
Similarly, the probability that A will not occur is
P共A⬘ 兲 ⫽
m
.
m⫹n
APPENDIX C.2
Example 8
Probability Computations
A15
Finding the Probability from the Odds
The odds of a horse winning a race are given as 2 to 1. Assuming these odds to
be correct, what is the probability of the horse winning? What is the probability
of the horse losing?
SOLUTION
Because the odds are n ⫽ 2 to m ⫽ 1, the probability of the horse
winning is
P共A兲 ⫽
n
2
2
⫽
⫽
m⫹n 1⫹2 3
and the probability of the horse losing is
P共A⬘ 兲 ⫽
m
1
1
⫽
⫽ .
m⫹n 1⫹2 3
✓CHECKPOINT 8
In Example 8, suppose the odds of the horse winning the race are 3 to 1. What
is the probability of the horse winning? losing? ■
CONCEPT CHECK
1. The ______ of an event A is the collection of all outcomes in the sample
space that are not in A.
P 冇A冈
2. If A is an event, then the ______ that A will occur are given by
,
P 冇A⬘ 冈
where P 冇A⬘ 冈 ⴝ 0.
3. Describe when two events A and B are mutually exclusive.
4. Describe the difference between the event A and A⬘.
A16
APPENDIX C
Skills Review C.2
Probability and Probability Distributions
The following warm-up exercises involve skills that were covered in earlier sections. You will use
these skills in the exercise set for this section. For additional help, review Appendix C.1.
In Exercises 1–2, determine the sample space for the given experiment.
1. A letter from the word probability is selected.
2. A card is selected from the set of black cards from a deck of playing cards.
In Exercises 3–4, two six-sided dice are tossed. Find the indicated probability.
3. The probability that the sum is 5.
4. The probability that the sum is less than 9.
Exercises C.2
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
In Exercises 1–4, a card is drawn from a standard deck
of 52 cards.
14. Blood Type In Exercise 13, what is the probability that
the patient chosen has type A, type B, or type AB blood?
1. What is the probability that the card is a face card or a 7?
15. Promotional Contest The number of customers who
entered a week-long contest is given in the following table.
What is the probability that the winner of the contest
entered on Sunday or Wednesday?
2. What is the probability that the card is a queen or an ace?
3. What is the probability that the card is a club or a diamond?
4. What is the probability that the card is a 2, 3, or 7?
Sunday
24
Thursday
61
Monday
32
Friday
32
Tuesday
29
Saturday
32
Wednesday
42
In Exercises 5–10, two six-sided dice are tossed.
5. What is the probability that the sum of the two numbers is
3 or 4?
6. What is the probability that the sum of the two numbers is
5 or 7?
7. What is the probability that the sum of the two numbers is
less than 8?
8. What is the probability that the sum of the two numbers is
greater than 4?
9. What is the probability that the product of the two numbers
is 2 or 3?
10. What is the probability that the product of the two numbers
is 2, 3, or 7?
11. State Lottery In a state lottery, a random number is
selected between 000 and 999. What is the probability that
the number is less than 100 or greater than 800?
12. Birthday If a birthdate is selected at random from
among 365 days, what is the probability that it is a date in
May or June?
13. Blood Type In a sample of 24 hospital patients, 9 of
them have type O blood, 6 have type A, 5 have type B, and
4 have type AB. If a patient is chosen at random from the
sample, what is the probability that the patient’s blood is
type A or type B?
16. Promotional Contest Use the table in Exercise 15 to
find the probability that the winner entered on Monday or
Saturday.
17. Find P共A 傼 B兲 if P共A兲 ⫽ 0.2, P共B兲 ⫽ 0.5 and
P共A 傽 B兲 ⫽ 0.1.
18. Find P共A 傽 B兲 if P 共A兲 ⫽ 1兾2, P共B兲 ⫽ 2兾3 and
P共A 傼 B兲 ⫽ 5兾6.
19. Find P共A兲 if P共B兲 ⫽ 1兾3, P共A 傼 B兲 ⫽ 2兾3 and
P共A 傽 B兲 ⫽ 1兾9.
20. Find P共B兲 if P共A兲 ⫽ 0.5, P共A 傼 B兲 ⫽ 0.8 and
P共A 傽 B兲 ⫽ 0.
Card Game In Exercises 21–24, one card is selected
from a standard deck of 52 playing cards. Find the
indicated probability.
21. The probability that the card is either a face card or a club.
22. The probability that the card is an ace or a diamond.
23. The probability that the card is black or a face card.
24. The probability that the card is red or a king.
APPENDIX C.2
Dice In Exercises 25–28, two six-sided dice are tossed.
Find the indicated probability.
25. The probability that the sum is odd or more than 8.
26. The probability that the sum is even or prime.
27. The probability that the product is 6 or the sum is 7.
28. The probability that the product is 3 or the sum is even.
29. Telephone Number If a computer generates the first
digit of a telephone number at random, what is the probability that the digit is even or less than 4?
30. Election Poll Three people have been nominated for
president of a college class. From a small poll, it is estimated that the probability of Candidate I winning the election
is 0.37, and the probability of Candidate II winning the
election is 0.44. What is the probability that the third candidate will win the election?
31. Random Selection A boat builder assembles sailboats
and motorboats at four locations (A, B, C, D) as shown in
the table. What is the probability that a boat selected at
random is a motorboat or was assembled at location D?
A
B
C
D
Sailboats
16
9
4
10
Motorboats
10
7
9
5
Probability Computations
A17
36. Random Selection A dormitory has 60 students, 24 of
whom are taking an English test, and 16 of whom are taking a math test. Of the students taking tests, 8 are taking
both tests. What is the probability that a student chosen at
random from the dormitory is taking a test?
37. Defective Units The probability of getting at least one
defective unit in a shipment of ten units is p. What is the
probability that all ten are nondefective?
38. Defective Units The probability of getting at least two
defective units in a shipment of ten units is 0.30. What is
the probability that at least nine of the units are nondefective?
39. Probability of a Complement The management of a
company plans to select one of its employees at random to
feature in the company’s next quarterly report. The probability of selecting a female employee is 2兾9. What is the
probability of selecting a male employee?
40. Probability of a Complement An auditorium is filled
with 500 people, 150 of whom are 40 years old or older.
What is the probability that someone selected at random
from the auditorium is under 40 years old?
41. Probability of a Complement If the only way to win
a card game is to select a face card or an ace from a standard deck of cards, what is the probability of losing the
game?
32. Random Selection Use the table in Exercise 31 to find
the probability that a boat selected at random is a sailboat
or was assembled at location B.
Random Selection In Exercises 33 and 34, a single
ball is drawn at random from a container filled with
50 balls, 25 red numbered from 1–25 and 25 black
numbered from 1–25.
33. What is the probability that a red or even-numbered ball is
drawn?
34. What is the probability that a black ball or a ball numbered
less than 10 is drawn?
35. Random Selection A section of a forest is composed
of several species of trees as shown in the table. What is the
probability that a tree cut down at random is a height that
is less than 50 feet or is an oak?
42. Probability of a Complement There are 48 cellphones at a phone store, 20 of which are flip phones. What
is the probability that a cell phone chosen at random is not
a flip phone?
In Exercises 43–46, find the odds of an event with the
given probability.
43. P共A兲 ⫽ 16
44. P共A兲 ⫽ 12
45. P共A兲 ⫽ 25
3
46. P共A兲 ⫽ 10
47. What are the odds of guessing the month of a stranger’s
birthdate?
48. What are the odds of drawing a club or a red card from a
standard deck of 52 playing cards?
In Exercises 49 and 50, the odds that an event will
occur are given. Find the probability (a) that the event
will occur, and (b) that the event will not occur.
49. 4 to 1
Height
Pine
Oak
Birch
Less than 50 feet
17
21
10
50 feet or more
20
16
16
50. 3 to 2
51. Probability of Winning A statistician figures that the
odds of a horse winning a race are 2 to 5. Based on these
odds, what is the probability that the horse will win the
race?
52. Probability of Winning The odds of the home team
winning a football game are 1 to 3. What is the probability
that the visiting team will win?