APPENDIX C.1
C
Probability
A1
Probability and Probability
Distributions
C.1
Probability
■ Find the sample space of an experiment.
■ Find the probability of an event.
Sample Space of an Experiment
When assigning measurements to the uncertainties of everyday life, people often
use ambiguous terminology such as “fairly certain,” “probable,” and “highly
unlikely.” Probability theory allows you to remove this ambiguity by assigning a
number to the likelihood of the occurrence of an event. This number is called the
probability that the event will occur. For example, if you toss a fair coin, the
probability that it will land heads up is one-half or 0.5.
In probability theory, any happening whose result is uncertain is called an
experiment. Each repetition of an experiment is called a trial. The possible
results of the experiment are outcomes, the set of all possible outcomes of the
experiment is the sample space of the experiment, and any subcollection of a
sample space is an event.
For instance, when a six-sided die is tossed, the sample space can be represented by the numbers 1 through 6. For this experiment, each of the outcomes is
equally likely.
To describe sample spaces in such a way that each outcome is equally
likely, you must sometimes distinguish between various outcomes in ways that
appear artificial. Example 1 illustrates such a situation.
Example 1
Finding Sample Spaces
Find the sample space for each of the following.
a. One coin is tossed.
b. Two coins are tossed.
SOLUTION
a. Because the coin will land heads up (denoted by H) or tails up (denoted by T),
the sample space is S ⫽ 再H, T冎.
b. Because either coin can land heads up or tails up, the possible outcomes are as
follows.
✓CHECKPOINT 1
An experiment consists of tossing
a coin and a six-sided die. Find the
sample space for the experiment. ■
HH ⫽ heads up on both coins
HT ⫽ heads up on first coin and tails up on second coin
TH ⫽ tails up on first coin and heads up on second coin
TT ⫽ tails up on both coins
So, the sample space is S ⫽ 再HH, HT, TH, TT冎. Note that the list distinguishes between the two cases HT and TH, even though these two outcomes appear
to be similar.
A2
APPENDIX C
Probability and Probability Distributions
The Probability of an Event
To calculate the probability of an event, count the number of outcomes in the event
and in the sample space. The number of equally likely outcomes in event E is
denoted by n共E兲, and the number of equally likely outcomes in the sample space
S is denoted by n共S兲. The probability that event E will occur is given by n共E兲兾n共S兲.
The Probability of an Event
If an event E has n共E兲 equally likely outcomes and its sample space S has
n共S兲 equally likely outcomes, then the probability of event E is
P共E兲 ⫽
Increasing likelihood
of occurrence
0.0
0.5
1.0
Impossible
event
(cannot
occur)
The occurrence
of the event is
just as likely as
it is unlikely.
Certain
event
(must
occur)
FIGURE C.1
n共E兲
.
n共S兲
Because the number of outcomes in an event must be less than or equal to the
number of outcomes in the sample space, the probability of an event must be a
number between 0 and 1. That is,
0 ≤ P共E兲 ≤ 1
as indicated in Figure C.1. If P共E兲 ⫽ 0, event E cannot occur, and E is called an
impossible event. If P共E兲 ⫽ 1, event E must occur, and E is called a certain
event.
Example 2
Finding Probabilities of Events
Find the probability of the following events.
a. Two coins are tossed. What is the probability that both land heads up?
b. A card is drawn from a standard deck of playing cards. What is the probability that it is an ace?
SOLUTION
STUDY TIP
You can write a probability as
a fraction, decimal, or percent.
For instance, in Example 2(a),
the probability of getting two
heads can be written as 14, 0.25,
or 25%.
a. Following the procedure in Example 1(b), let
E ⫽ 再HH冎 and S ⫽ 再HH, HT, TH, TT 冎.
The probability of getting two heads is
P共E兲 ⫽
n共E兲 1
⫽ .
n共S兲
4
b. Because there are 52 cards in a standard deck of playing cards and there are
4 aces (one in each suit), the probability of drawing an ace is
P共E兲 ⫽
n共E兲
4
1
⫽
⫽ .
n共S兲
52 13
✓CHECKPOINT 2
A card is drawn from a standard deck of playing cards. What is the probability
that it is a face card (king, queen, or jack)? ■
APPENDIX C.1
Example 3
Probability
A3
Finding the Probability of an Event
Two six-sided dice are tossed. What is the probability that the sum of the dice is
7? (See Figure C.2.)
SOLUTION Because there are 6 possible outcomes on each die, you can use the
Fundamental Counting Principle to conclude that there are 6 ⭈ 6 or 36 different
outcomes when two dice are tossed. To find the probability of rolling a sum of 7,
you must first count the number of ways in which this can occur.
FIGURE C.2
First die
1
2
3
4
5
6
Second die
6
5
4
3
2
1
So, a total of 7 can be rolled in 6 ways, which means that the probability of rolling
a 7 is
Algebra Review
For examples on how to count
the number of ways an event
can happen, see the Chapter 11
Algebra Review on pages 691
and 692.
P共E兲 ⫽
n共E兲
6
1
⫽
⫽ .
n共S兲
36 6
✓CHECKPOINT 3
One six-sided die is tossed twice. What is the probability that the sum of the
two tosses is 4? ■
Example 4
Finding the Probability of an Event
Twelve-sided dice, as shown in Figure C.3, can be constructed (in the shape of
regular dodecahedrons) such that each of the numbers from 1 to 6 appears twice
on each die. Show that these dice can be used in any game requiring ordinary sixsided dice without changing the probabilities of different outcomes.
SOLUTION For an ordinary six-sided die, each of the numbers 1, 2, 3, 4, 5, and
6 occurs only once, so the probability of any particular number is
P共E兲 ⫽
FIGURE C.3
n共E兲 1
⫽ .
n共S兲
6
For one of the twelve-sided dice, each number occurs twice, so the probability of
any particular number is
P共E兲 ⫽
n共E兲
2
1
⫽
⫽ .
n共S兲
12 6
✓CHECKPOINT 4
One twelve-sided die is tossed twice. What is the probability that the sum of the
two tosses is 11? ■
A4
APPENDIX C
Probability and Probability Distributions
Example 5
The Probability of Winning a Lottery
In a state lottery, a player chooses 6 different numbers from 1 to 40. If these six
numbers match the six numbers drawn (in any order) by the lottery commission,
the player wins (or shares) the top prize. What is the probability of winning the
top prize if the player buys one ticket?
SOLUTION Because the order of the numbers is not important, use the formula
for the number of combinations of 40 elements taken 6 at a time to determine the
size of the sample space.
n共S兲 ⫽ 40C6 ⫽
40
⭈ 39 ⭈ 38 ⭈ 37 ⭈ 36 ⭈ 35 ⫽ 3,838,380
6⭈5⭈4⭈3⭈2⭈1
If a person buys only one ticket, the probability of winning the top prize is
P共E兲 ⫽
n共E兲
1
⫽
.
n共S兲
3,838,380
✓CHECKPOINT 5
A bag contains two green, three yellow, and four red marbles. If two marbles
are drawn from the bag without replacement, what is the probability that both
marbles are red? ■
CONCEPT CHECK
1. What is an experiment?
2. What is a sample space?
n 冇E冈
,
n 冇S冈
where n 冇E冈 is the number of equally likely outcomes in the event and n 冇S冈
is the number of equally likely outcomes in the sample space.
3. To determine the ______ of an event, you can use the formula P 冇E冈 ⴝ
4. If P 冇E冈 ⴝ 0, then E is a(n) ______ event, and if P 冇E冈 ⴝ 1, then E is
a(n) ______ event.
APPENDIX C.1
Skills Review C.1
Probability
A5
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 the Chapter 11
Algebra Review on pages 691 and 692.
In In Exercises 1– 4, write the fraction (a) in lowest terms, (b) as a decimal,
and (c) as a percent.
1.
6
8
2.
13
52
3.
230
500
4.
x
10x
In Exercises 5–10, evaluate the expression.
5.
8.
1
6.
8C4
9
⭈8⭈7⭈6⭈5
9!
9.
1
6!
7.
6C3
10.
10C3
Exercises C.1
In Exercises 1–4, determine the sample space for the
experiment.
⭈4
5
5!
16C2
⭈ 4C3
20C5
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Tossing a Die In Exercises 13–18, find the probability
for the experiment of tossing a six-sided die twice.
1. A six-sided die is tossed twice and the sum is recorded.
13. The sum is 6.
2. A taste tester has to rank three varieties of yogurt, A, B, and
C, according to preference.
14. The sum is at least 7.
3. Two marbles are selected from a bag containing two red
marbles, two blue marbles, and one yellow marble. The
color of each marble is recorded.
16. The sum is 2, 3, or 12.
4. Two county supervisors are selected from five supervisors,
A, B, C, D and E, to study a recycling plan.
18. The sum is odd or prime.
Tossing a Coin In Exercises 5–8, find the probability
for the experiment of tossing a coin three times. Use
the sample space S ⴝ 再 HHH, HHT, HTH, HTT, THH, THT,
TTH, TTT冎.
5. The probability of getting exactly one tail
6. The probability of getting a head on the first toss
15. The sum is less than 11.
17. The sum is no more than 7.
In Exercises 19–22, a computer generates an integer
from 1 through 20 at random. Find the probability of
the event.
19. The probability of generating a multiple of 4
20. The probability of generating a number divisible by 5
21. The probability of generating a prime number
22. The probability of generating a factor of 24
7. The probability of getting at least one head
8. The probability of getting at least two heads
Drawing a Card In Exercises 9–12, find the probability for the experiment of selecting 1 card from a standard deck of 52 playing cards.
9. The card is black.
Drawing Marbles In Exercises 23–26, find the probability for the experiment of drawing two marbles
(without replacement) from a bag containing one
green, two yellow, and three red marbles.
23. Both marbles are red.
24. Both marbles are yellow.
10. The card is a red face card.
25. Neither marble is yellow.
11. The card is a 6 or lower. (Aces are low.)
26. Neither marble is red.
12. The card is not a face card.
A6
APPENDIX C
Probability and Probability Distributions
27. Jury Selection A person is selected at random for jury
duty from a list of registered voters from three different
counties. Country A has 14,789 registered voters, County B
has 17,851 registered voters, and County C has 23,487 registered voters. If only one name is selected, what is the
probability that the person chosen is from County C?
28. Order of Arrival Three fire engines, four police cars,
and one ambulance are called to the scene of an accident. If
they all have an equal chance of arriving at the same time,
what is the probability that a police car will arrive first?
29. Random Selection Nine players went to bat in the
sixth inning of a baseball game. Four had singles, one had
a double, one had a grand slam, and the others struck out.
What is the probability that a batter chosen at random
struck out?
30. Contract Bidding Ten health insurance companies
are bidding for an insurance contract. Three are local
companies, three have state-wide operations, and four companies have national operations. Only one company will be
awarded the contract. If each company is equally likely to
win the contract, what is the probability that the contract
will be awarded to one of the companies with state-wide
operations?
31. Data Analysis A study of the effectiveness of a flu
vaccine was conducted with a sample of 500 people. Some
participants in the study were given no vaccine, some were
given one injection, and some were given two injections.
The results of the study are listed in the table.
No
vaccine
One
injection
Two
injections
Total
7
2
13
22
No flu
149
52
277
478
Total
156
54
290
500
Flu
A person is selected at random from the sample. Find the
indicated probability
(a) The person had two injections.
(b) The person did not get the flu.
(c) The person got the flu and had one injection.
32. Data Analysis One hundred college students were
interviewed to determine their political party affiliations
and whether they favored a balanced-budget amendment
to the Constitution. The results of the study are listed in
the table, where D represents Democrat and R represents
Republican.
Favor
Not Favor
Unsure
Total
D
23
25
7
55
R
32
9
4
45
Total
55
34
11
100
A person is selected at random from the sample. Find the
probability that the person described is selected.
(a) A person who does not favor the amendment
(b) A Republican
(c) A Democrat who favors the amendment
33. Alumni Association A college sends a survey to
selected members of the class of 2009. Of the 1254 people
who graduated that year, 672 are women, of whom 124
went on to graduate school. Of the 582 male graduates, 198
went on to graduate school. An alumni member is selected
at random. What are the probabilities that the person is
(a) female, (b) male, and (c) female and did not attend
graduate school?
34. Education In a high school graduating class of 202 students, 95 are on the honor roll. Of these, 71 are going on to
college, and of the other 107 students, 53 are going on to
college. A student is selected at random from the class.
What are the probabilities that the person chosen is (a) on
the honor roll, (b) going to college, and (c) on the honor
roll, but not going to college?
35. Defective Item A clerk sold an equal number of hats,
scarves, gloves, ski masks, and ear muffs. If one of the
items was returned because it was defective, what is the
probability that it was a hat?
36. Birth Order Each of six mothers-to-be received 3D
ultrasound scans, which showed that four of them will give
birth to girls. What is the probability that the first two
women to give birth will have boys?
37. Random Selection Four letters and envelopes are
addressed to four different people. If the letters are inserted into the envelopes at random, what is the probability that
exactly one letter will be inserted in the correct envelope?
38. Random Selection A math teacher chooses 5 students
at random from a class of 20 to solve a problem at the
board. If 12 students know how to solve the problem, what
is the probability that (a) all 5 students picked do not know
how to solve the problem, and (b) exactly 3 students picked
know how to solve the problem?
39. Defective Units A shipment of 12 microwave ovens
contains 3 defective units. A vending company has ordered
4 of these units. Because the microwave ovens are identically packaged, the selection will be at random. What is the
probability that (a) all 4 units are good, and (b) exactly 2
units are good?