Appendix G
G1
Probability
Appendix G Probability
What You Should Learn
1 Determine the probability that an event will occur.
2 Use counting principles to determine the probability that an event will occur.
Why You Should Learn It
Counting methods can be used to determine
probabilities of real-life events. For instance, in
Exercise 44 on page G8, you can determine the
probability of having a winning lottery ticket.
1 Determine the probability that an
event will occur.
The Probability of an Event
The probability of an event is a number from 0 to 1 that indicates the likelihood
that the event will occur. An event that is certain to occur has a probability of 1.
An event that cannot occur has a probability of 0. An event that is equally likely
1
to occur or not occur has a probability of 2 or 0.5.
Probability of 0:
Event cannot
occur.
Probability of 0.5:
Event is equally likely
to occur or not occur.
Probability of 1:
Event is certain
to occur.
0
0.5
1
The Probability of an Event
Consider a sample space S that is composed of a finite number of outcomes, each of which is equally likely to occur. A subset E of the sample
space is an event. The probability that an outcome E will occur is
P
Number of outcomes in event
.
Number of outcomes in sample space
Example 1 Finding the Probability of an Event
a. You select a card from a standard deck of 52 cards. The probability that the
card is an ace is
P
Number of aces in the deck
4
1
.
Number of cards in the deck 52 13
b. On a multiple-choice test, you know that the answer to a question is not (a) or
(d), but you are not sure about (b), (c), and (e). If you guess, the probability
that you are wrong is
P
Number of wrong answers
2
.
Number of possible answers 3
G2
Appendix G
Probability
Example 2 Conducting a Poll
The Centers for Disease Control surveyed 11,631 high school students. The
students were asked whether they considered themselves to be a good weight,
underweight, or overweight. The results are shown in Figure G.1.
Female
Good weight
3082
Male
Overweight
1776
Underweight
366
Good weight
4389
Overweight
961
Underweight
1057
Figure G.1
a. You choose a female at random from those surveyed. The probability that she
said she was underweight is
P
Number of females who answered “underweight”
Number of females in survey
366
3082 366 1776
366
5224
0.07.
b. You choose a person who answered “underweight” from those surveyed. The
probability that the person is female is
P
Number of females who answered “underweight”
Number in survey who answered “underweight”
366
366 1057
366
1423
0.26.
Polls such as the one described in Example 2 are often used to make inferences about a population that is larger than the sample. For instance, from
Example 2, you might infer that 7% of all high school girls consider themselves
to be underweight. When you make such an inference, it is important that those
surveyed are representative of the entire population.
Appendix G
Probability
G3
Example 3 Using Area to Find Probability
You have just stepped into the tub to take a shower when one of your contact lenses falls out. (You have not yet turned on the water.) Assuming the lens is equally
likely to land anywhere on the bottom of the tub, what is the probability that it
lands in the drain? Use the dimensions shown in Figure G.2.
Drain
26 in.
} 2 in.
50 in.
Figure G.2
Solution
Because the area of the tub bottom is
2650 1300
Area of tub width length
square inches and the area of the drain is
12 Area of drain r 2
square inches, the probability that the lens lands in the drain is
P
0.0024.
1300
Example 4 The Probability of Inheriting Certain Genes
Common parakeets have genes that can produce any one of four feather colors.
BC
Bc
bC
bc
BC BBCC BBCc BbCC BbCc
Bc BBCc BBcc
BbCc
Bbcc
Solution
The probability that an offspring will be yellow is
bbCC
bbCc
bc BbCc
bbCc
bbcc
Figure G.3
BBCC, BBCc, BbCC, BbCc
BBcc, Bbcc
bbCC, bbCc
bbcc
Use the Punnett square shown in Figure G.3 to find the probability that an
offspring of two green parents (both with BbCc feather genes) will be yellow.
Note that each parent passes along a B or b gene and a C or c gene.
bC BbCC BbCc
Bbcc
Green:
Blue:
Yellow:
White:
P
Number of yellow possibilities
Number of possibilities
3
.
16
G4
Appendix G
Probability
2 Use counting principles to determine
the probability that an event will occur.
Using Counting Methods to Find Probabilities
In the following examples, you will see how the basic counting principles are
used to determine the probability that an event will occur.
Example 5 The Probability of a Royal Flush
Five cards are dealt at random from a standard deck of 52 playing cards (see Figure
G.4). What is the probability that the cards are 10-J-Q-K-A of the same suit?
Standard 52-Card Deck
A♠
K♠
Q♠
J♠
10♠
9♠
8♠
7♠
6♠
5♠
4♠
3♠
2♠
A♥
K♥
Q♥
J♥
10♥
9♥
8♥
7♥
6♥
5♥
4♥
3♥
2♥
A♦
K♦
Q♦
J♦
10♦
9♦
8♦
7♦
6♦
5♦
4♦
3♦
2♦
A♣
K♣
Solution
The number of possible five-card hands from a deck of 52 cards is
52C5 2,598,960. Because only four of these five-card hands are
10-J-Q-K-A of the same suit, the probability that the hand contains these cards is
Q♣
J ♣
10 ♣
9♣
8♣
P
♣
♣
5♣
4♣
3♣
2♣
7
6
Example 6 Conducting a Survey
A survey was conducted of 500 adults who had worn Halloween costumes. Each
person was asked how he or she acquired a Halloween costume: created it, rented
it, bought it, or borrowed it. The results are shown in Figure G.5. What is the
probability that the first four people who were polled all created their costumes?
Figure G.4
Created
360
Borrowed
20
Rented
60
Bought
60
Figure G.5
4
1
.
2,598,960 649,740
Solution
To answer this question, you need to use the formula for the number of combinations twice. First, find the number of ways to choose four people from the 360
who created their own costumes.
360C4
360
359 358 357 688,235,310
4321
Next, find the number of ways to choose four people from the 500 who were
surveyed.
500C4
500
499 498 497 2,573,031,125
4321
The probability that all of the first four people surveyed created their own
costumes is the ratio of these two numbers.
P
Number of ways to choose 4 from 360
Number of ways to choose 4 from 500
688,235,310
2,573,031,125
0.267
Appendix G
Probability
G5
Appendix G Exercises
Review Concepts, Skills, and Problem Solving
Keep mathematically in shape by doing these exercises
before the problems of this section.
9. ln
7
x3
10. ln
uu 22
2
Properties and Definitions
In Exercises 1– 6, complete the property of logarithms.
Graphs and Models
log a1 log a a x
log a a log auv u
5. log a v
6. log a un 11.
1.
2.
3.
4.
y
10,000
.
1 4ex3
(a) Use a graphing calculator to graph the equation.
Rewriting Expressions
In Exercises 7–10, use the properties of logarithms to
expand the expression as a sum, difference, or multiple
of logarithms.
7. log2x2y
Sales Annual sales y of a product x years after
it is introduced are approximated by
8. log5xx2 1
(b) Use the graph in part (a) to approximate the time
when annual sales are y 5000 units.
(c) Use the graph in part (a) to estimate the maximum level of annual sales.
12. Investment An investment is made in an account
that pays 5.5% interest, compounded continuously.
What is the effective yield of this investment?
Explaining Concepts
Sample Space In Exercises 1–4, determine the number
of outcomes in the sample space for the experiment.
1. One letter from the alphabet is chosen.
2. A six-sided die is tossed twice and the sum is recorded.
3. Two county supervisors are selected from five supervisors, A, B, C, D, and E, to study a recycling plan.
4. A salesperson makes product presentations in three
homes per day. In each home there may be a sale
(denote by Y) or there may be no sale (denote by N).
Sample Space In Exercises 5– 8, list the outcomes in
the sample space for the experiment.
7. A basketball tournament between two teams consists
of three games. For each game, your team may win
(denote by W) or lose (denote by L).
8. Two students are randomly selected from four
students, A, B, C, and D.
In Exercises 9 and 10, you are given the probability that
an event will occur. Find the probability that the event
will not occur.
9. p 0.35
10. p 0.80
5. A taste tester must taste and rank three brands of
yogurt, A, B, and C, according to preference.
In Exercises 11 and 12, you are given the probability
that an event will not occur. Find the probability that
the event will occur.
6. A coin and a die are tossed.
11. p 0.82
12. p 1.00
G6
Appendix G
Probability
Coin Tossing In Exercises 13–16, a coin is tossed three
times. Find the probability of the specified event. Use
the sample space
S HHH, HHT, HTH, HT T, THH, THT, T TH, T T T.
See Example 1.
13.
14.
15.
16.
The event of getting exactly two heads
The event of getting a tail on the second toss
The event of getting at least one head
The event of getting no more than two heads
Playing Cards In Exercises 17–20, a card is drawn
from a standard deck of 52 playing cards. Find the
probability of drawing the indicated card.
17. A red card
19. A face card
18. A queen
20. A black face card
Tossing a Die In Exercises 21–24, a six-sided die is
tossed. Find the probability of the specified event.
21.
22.
23.
24.
The number is a 5.
The number is a 7.
The number is no more than 5.
The number is at least 1.
United States Blood Types In Exercises 25 and 26, use
the circle graph, which shows the percent of people in
the United States with each blood type. See Example 2.
(Source: American Red Cross)
Type B+ 9%
Type AB− 1%
Type A+
34%
Type AB+ 3%
Type A− 6%
Type O+
38%
Type B− 2%
Type O − 7%
25. A person is selected at random from the United
States population. What is the probability that the
person does not have blood type B?
26. What is the probability that a person selected at
random from the United States population does have
blood type B? How is this probability related to the
probability found in Exercise 25?
Family Incomes In Exercises 27–30, use the circle
graph, which shows the percent of families in the
United States that earned incomes in each given dollar
range for the year 2000. Answer the question for a
family selected at random from the population.
(Source: U.S. Census Bureau)
Under $15,000
9.5%
$15,000–$24,999
11.5%
$25,000–$34,999
12.0%
$35,000–$49,999
15.9%
$100,000 and over
17.0%
$75,000–$99,999
12.6%
$50,000–$74,999
21.5%
27. What is the probability that the family earned at least
$100,000?
28. What is the probability that the family earned less
than $15,000?
29. What is the probability that the family earned less
than $50,000?
30. What is the probability that the family earned at least
$25,000?
31. Multiple-Choice Test A student takes a multiplechoice test in which there are five choices for each
question. Find the probability that the first question
is answered correctly given the following conditions.
(a) The student has no idea of the answer and
guesses at random.
(b) The student can eliminate two of the choices and
guesses from the remaining choices.
(c) The student knows the answer.
32. Multiple-Choice Test A student takes a multiplechoice test in which there are four choices for each
question. Find the probability that the first question
is answered correctly given the following conditions.
(a) The student has no idea of the answer and
guesses at random.
(b) The student can eliminate two of the choices and
guesses from the remaining choices.
(c) The student knows the answer.
33. Class Election Three people are running for class
president. It is estimated that the probability that
candidate A will win is 0.5 and the probability that
candidate B will win is 0.3. What is the probability
that either candidate A or candidate B will win?
Geometry In Exercises 35– 38, the specified
probability is the ratio of two areas. See Example 3.
35. Meteorites The largest meteorite in the United
States was found in the Willamette Valley of Oregon
in 1902. Earth contains 57,510,000 square miles of
land and 139,440,000 square miles of water. What is
the probability that a meteorite that hits the Earth
will fall onto land? What is the probability that a
meteorite that hits the Earth will fall into water?
36. Game A child uses a spring-loaded device to shoot
a marble into the square box shown in the figure. The
base of the box is horizontal and the marble has an
equal likelihood of coming to rest at any point in the
box. In parts (a)–(d), find the probability that the
marble comes to rest in the specified region.
(a) The red center
(b) The blue ring
(c) The purple border (d) Not in the yellow ring
2 2 2 2 2
Probability
You meet
You meet
You don't meet
30
15
Yo
u
ar
rf
riv
rie
ef
nd
irs
ar
t
riv
es
fir
st
60
45
G7
Yo
u
34. Winning an Election Jones, Smith, and Thomas
are candidates for public office. It is estimated that
Jones and Smith have about the same probability
of winning, and Thomas is believed to be twice as
likely to win as either of the others. Find the probability of each candidate winning the election.
Your friend’s arrival time
(in minutes past 5:00 P.M.)
Appendix G
15
30
45
60
Your arrival time
(in minutes past 5:00 P.M.)
Figure for 37
38. Estimating A coin of diameter d is dropped onto
a paper that contains a grid of squares d units on a
side (see figure).
(a) Find the probability that the coin covers a vertex
of one of the squares in the grid.
(b) Repeat the experiment 100 times and use the
results to approximate .
d
d
d
24 in
In Exercises 39 and 40, complete the Punnett square
and answer the questions. See Example 4.
24 in.
37. Meeting Time You and a friend agree to meet at a
favorite restaurant between 5:00 and 6:00 P.M. The
one who arrives first will wait 15 minutes for the
other, after which the first person will leave (see
figure). What is the probability that the two of you
actually meet, assuming that your arrival times are
random within the hour?
39. Girl or Boy? The genes that determine the genders
of humans are denoted by XX (female) and XY
(male). (See figure on next page.)
(a) What is the probability that a newborn will be a
girl? a boy?
(b) Explain why it is equally likely that a newborn
baby will be a boy or a girl.
G8
Appendix G
Probability
A
o
B
?
?
o
?
?
Female
X
X
?
?
Y
?
?
Male
X
Figure for 39
Figure for 40
40. Blood Types There are four basic human blood
types: A (AA or Ao), B (BB or Bo), AB (AB), and O
(oo).
(a) What is the blood type of each parent in the figure?
(b) What is the probability that their offspring will
have blood type A? B? AB? O?
In Exercises 41–50, some of the sample spaces are large.
Therefore, you should use the counting principles
discussed in Section 11.5. See Examples 5 and 6.
41. Game Show On a game show, you are given five
digits to arrange in the proper order to form the price
of a car. If you arrange them correctly, you win the
car. Find the probability of winning if you know the
correct position of only one digit and must guess the
positions of the other digits.
42. Game Show On a game show, you are given four
digits to arrange in the proper order to form the price
of a grandfather clock. What is the probability of
winning if
(a) you guess the position of each digit?
(b) you know the first digit, but must guess the
remaining three?
43. Shelving Books A parent instructs a young child
to place a five-volume set of books on a bookshelf.
Find the probability that the books are in the correct
order if the child places them at random.
44. Lottery You buy a lottery ticket inscribed with a
five-digit number. On the designated day, five digits
(from 0 to 9, inclusive) are randomly selected. What
is the probability that you have a winning ticket?
45. Preparing for a Test An instructor gives her class
a list of 10 study problems, from which she will
select eight to be answered on an exam. You know
how to solve eight of the problems. What is the
probability that you will be able to answer all eight
questions on the exam correctly?
46. Committee Selection A committee of three
students is to be selected from a group of three girls
and five boys. Find the probability that the
committee is composed entirely of girls.
47. Defective Units A shipment of 10 food processors
to a certain store contains two defective units. You
purchase two of these food processors as birthday
gifts for friends. Determine the probability that you
get both defective units.
48. Defective Units A shipment of 12 compact disc
players contains two defective units. A husband and
wife buy three of these compact disc players to give
to their children as Christmas gifts.
(a) What is the probability that none of the three
units is defective?
(b) What is the probability that at least one of the
units is defective?
49. Card Selection Five cards are selected from a
standard deck of 52 cards. Find the probability that
all hearts are selected.
50. Card Selection Five cards are selected from a
standard deck of 52 cards. Find the probability that
two aces and three queens are selected.
Explaining Concepts
51. The probability of an event must be a real number in
what interval? Is the interval open or closed?
52.
The probability of an event is 34. What is
the probability that the event does not occur? Explain.
53.
What is the sum of the probabilities of
all the occurrences of outcomes in a sample space?
Explain.
54.
The weather forecast indicates that the
probability of rain is 40%. Explain what this means.