1. science
2. computer science
3. artificial intelligence

Chapter 27
Quantifying Uncertainty
Chance and related terms such as likelihood, odds, and the more precise probability,
are ideas that apply to many situations involving uncertainty. We know that some
events are more likely to occur than others. But what does more likely mean? What is
the chance that a baby will be female? What are the chances of rain this week? What
is the chance that a taxi will arrive within 5 minutes? What is a candidate’s chance of
winning an election?
Notes
Recall from Chapter 1 that a quantity is anything (an object, event, or quality thereof)
that can be measured or counted and to which we can assign a value. In this chapter
we explore ways of assigning values as measures of uncertainty along with the basic
vocabulary for probability (the usual mathematical term for chance and uncertainty).
In particular, we discuss two basic methods of assigning numerical values to
probabilities.
27.1 Understanding Chance Events
In this section we focus on understanding and quantifying chance. We first consider
what is and what is not a probabilistic situation.
Many of us have experienced, at some time, events that were entirely unexpected.
Consider the following coincidences.
1. Cassie and Rachelle were roommates at New England University. They met unexpectedly 10 years later at Old Faithful in Yellowstone Park. Cassie asked,
“What are the chances of running into each other here?”
2. Rishad’s sister, Betty, rolled 10 sixes in a row while playing a board game.
Rishad said, “That’s impossible!”
Statements and questions such as these are common. But in terms of understanding
chance events they are actually not appropriate because they refer to the chances of
an event that has already happened and for which the outcome is already known.
Either the event happened or it did not. Either the two former roommates met at
Old Faithful or they did not. And Betty either rolled 10 sixes or she did not.
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Notes
Understanding why the statements and questions are inappropriate is important, and
this underlying idea will recur throughout the chapters dealing with chance.
In the case of the two New England University roommates meeting at Old Faithful,
the chances are certain that this could happen because Cassie and Rachelle did meet.
In the case of the board game, Rishad’s sister did throw 10 sixes in a row, so it certainly is possible that she could.
Now consider some reformulations of the same two situations. Compare these alternate statements and questions to the original ones.
1A.
Cassie and Rachelle were roommates at New England University. They met
unexpectedly 10 years later at Old Faithful in Yellowstone Park. Cassie said,
“What a coincidence! I wonder what fraction of pairs of roommates from New
England University accidentally meet far away from either one’s home 10
years after graduating?”
2A. Rishad’s sister, Betty, rolled 10 sixes in a row while playing a board game.
Rishad asked, “If a billion people each rolled a die 10 times, I wonder what
fraction of those people might get sixes on every roll?”
Discussion 1
What’s the Difference?
What is different about the two versions of Cassie’s question? Of Rishad’s question?
There is an important difference between the original situations and their corresponding reformulations. The revised questions do not refer to a specific event that has already happened on a specific occasion. Rather, they presume that some process will
be repeated. Cassie’s revised question presumes that we repeatedly examine pairs of
roommates from New England University to see if they met in an out-of-the-way
place within 10 years of graduating. Rishad’s revised question moves the focus away
from Betty’s accomplishment as an isolated occurrence and asks how common her
accomplishment would be in a large number of similar situations.
We have already used some terms that need to be defined before we continue.
A probabilistic situation is a situation in which we are interested in the fraction
of the number of repetitions of a particular process that produces a particular
result when repeated under identical circumstances a large number of times.
In each of the revised descriptions of the two situations, one involving a chance
meeting and the other involving a random roll of a pair of dice, some process was
repeated a large number of times. Thus each describes a probabilistic situation. The
first descriptions, however, do not describe probabilistic situations because in each
case the event happened, that is, the two friends did meet, and Betty did roll 10 sixes
in a row. No process was involved in either of the two initial descriptions.
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Section 27.1 Understanding Chance Events
Notes
A process undertaken a large number of times, together with the results, is often
called an experiment. An outcome is a result of an experiment.
EXAMPLE 1
The rolling of a pair of dice a large number of times, and noting the number of dots
on top of each die, is a probabilistic situation because the rolling of the dice over and
over again is done the same way. Carrying out this process of rolling dice many
times and noting the numbers of dots on the top of the dice is an experiment. Each
time the dice are rolled, some numbers of dots appear on top, say 3 on one and 5 on
the other. This outcome gives a sum of 8.
An event is an outcome or the set of outcomes of a designated type. An event’s
probability is the fraction of the times the event will occur when some process is
repeated a large number of times.
EXAMPLE
2
Continuing from Example 1, a roll of dice resulting in a 4 and a 6 is also an outcome,
but this time the sum is 10. All possible outcomes that would give a sum of 10, for
example, give an event. If the dice are red and black, then a 4 on red and 6 on black,
a 5 on red and a 5 on black, and a 6 on red and a 4 on black would be the three possible outcomes in the event of obtaining a 10 on a roll of the dice. Now suppose that
you undertake an experiment of tossing two dice 1000 times, and suppose that the
event of the dots on top adding to 10 occurs 82 times. We would say that, based on
82
this experiment, the probability of rolling a 10 is 1000
or 0.082.
Discussion 2
1.
2.
3.
4.
5.
Practice Using These Terms
Give an example of an experiment involving a penny and a dime.
Describe an outcome of this experiment.
Describe a specific event for this experiment.
How can your experiment be a probabilistic situation?
How would you find the probability of the event you described in Problem 3?
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Notes
Quantifying Uncertainty
It often happens that people are thinking of a probabilistic situation but pose questions as if they were thinking about the outcome of a single event. Consider the following questions and their rephrasings.
Question 1: You toss two coins. What is the probability that you get two heads?
Rephrase:
Suppose a large number of people toss two coins. What fraction of the
tosses will end up with two heads?
Rephrase:
Suppose one person tosses two coins a large number of times. What
fraction of the tosses will end up with two heads?
Question 2: What is the probability of drawing an ace from a standard deck of
playing cards?
Rephrase:
Suppose a large number of people draw a card from a standard deck of
playing cards (and then put it back). What fraction of the draws will
result in an ace?
Rephrase:
Suppose you draw a card from a standard deck (and put it back) a large
number of times. What fraction of the draws will result in an ace?
Each question was revised twice—once supposing that a large number of people
complete the same process one time and once supposing that one person completes
the same process a large number of times. These revisions highlight the fact that it is
the process (such as tossing two coins, or drawing a card from a deck) that we imagine being repeated. A probabilistic situation is one that can be repeated and is future
oriented. In a room without a window you can be uncertain about whether or not it is
raining at that moment, but this situation is not a probabilistic situation. Whether or
not it will rain tomorrow is a probabilistic situation.
Discussion 3
Rephrasing
Rephrase each question so that the situation describes a process being repeated
a large number of times.
1. What is the probability that on a toss of three coins you will get two or more
heads?
2. What is the probability that you draw a red ball out of an opaque bag that contains two red balls and three blue balls?
3. What is the probability that it will rain tomorrow?
T AKE-AWAY MESSAGE . . . The probability of an event is the fraction of the number of times
that the event will occur when some process is repeated a large number of times. The
term probability is often meant when chance, likelihood, or uncertainty are used. People
sometimes make statements that sound as though they are speaking of the outcome of a
single occasion. However, in many cases the situation is probabilistic and that they are actually interested in what happens when a process is repeated a large number of times. ♦
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Section 27.1 Understanding Chance Events
Notes
Learning Exercises for Section 27.1
1. For each of the following statements, say whether or not the situation is probabilistic, and explain why or why not. If not, rephrase the statement so that it
describes a probabilistic situation.
a. The probability that the U.S. Secretary of State is a woman.
b. The probability that a person will be able to read newspapers more intelligently after completing this course.
c. The probability that the soon-to-be-born baby of a pregnant Mrs. Johnson
with two sons will be a girl.
d. The probability that a woman’s planned third child will be a girl, given the
information that her first two children are boys.
e. The probability that it will snow tomorrow in this city.
f. The probability that Joe ate pizza yesterday.
2. Because a probability is a fraction of a number of repetitions of some process,
what is the least value a probability can have, as a percent? What is the greatest
value it can have, as a percent? Explain your thinking.
3. Give examples that clarify the distinction between an uncertain situation and a
probabilistic situation.
4. Rephrase each sentence to show that the writer is correctly thinking about a
probabilistic situation.
a. If I toss this coin right now, the probability of heads is 50%.
b. If I pick a workday at random, the probability that it will be Monday is
1
5
.
5. a. Describe a probabilistic situation involving three coins: a penny, a nickel,
and a dime.
b. Describe an experiment involving the three coins in this situation.
c. Describe an outcome of this experiment.
d. Suppose one specific event is getting two heads. Which outcomes would be
in this event? Which outcomes would not be in this event?
6. a. Describe an experiment involving a toss of three dice.
b. Describe an outcome of this experiment.
c. Describe an event related to the outcome of the experiment.
d. List the outcomes that would be in the event, a sum of 5 for the dots showing
on the dice.
7. a. Describe an experiment of the drawing of three cards from a deck of cards
from which the Jacks, Queens, and Kings have been removed. (Note: There
are 52 cards in a deck, 13 of which are hearts, 13 of which are diamond, 13
of which are clubs, and 13 of which are spades. A card with an ace counts as
1. Nine cards of each suit are marked 2 through 10. Ignore the Jacks, Queens,
and Kings, leaving 40 cards from which to draw.)
b. Is drawing a 3 of hearts, a 4 of diamonds, and a 5 of spades an outcome or an
event? Why?
c. Is the drawing of three cards whose numbers add to 10 an outcome or an
event? Why?
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Notes
27.2 Methods of Assigning Probabilities
Methods of assigning probabilities to events—determining what fraction of the time
we should anticipate something to happen—rely on proportional reasoning. If we
find that an event occurs 23 of the time in a particular situation, then we use proportional reasoning to expect the event to occur approximately
2
3
of the time for any
number of times. So if the situation happened 300 times, we would expect the event
to occur approximately 200 times.
There are two ways to assign probabilities: theoretically or experimentally. Events
for which a probability can be arrived at by knowledge based on a theory of what is
likely to occur in a situation, such as when a fair coin is tossed, is a theoretical
probability.
EXAMPLE 3
In the toss of one die, the possible outcomes are 1, 2, 3, 4, 5, and 6. Each is equally
likely. Thus the theoretical probability of obtaining a 2 is 16 . If a coin is tossed, the
theoretical probability of obtaining a head is
cal probability of getting a 3 is
1
4
1
2
. With the spinner below, the theoreti-
. The theoretical probability of getting a 2 is
1
1
2
.
3
2
An experimental (or empirical) probability is an application of the adage, “What
has happened in the past will happen in the future.” The experimental probability of
an event is determined by undertaking a process a large number of times and noting
the fraction of time the particular event occurs. Thus an experimental probability
value is determined by undertaking an experiment and observing what happens.
Probabilities determined this way will vary, but the variation diminishes as the number of trials increases.
Determining a probability via an experiment is necessary when we have no prior
knowledge of possible probabilities. The activities in this section will give you practice in assigning both experimental and theoretical probabilities.
Section 27.2
633
Methods of Assigning Probabilities
Notes
Activity 1
A Tackful Experiment
We want to know the probability of a thumbtack landing point up if it is tossed from
a paper cup.
a. Separate into groups. Within each group, toss a thumbtack onto the floor (or the
top of a desk or table) 50 times. Keep track of how many times the thumbtack
points up.
b. What would you say is the probability that if a thumbtack is tossed from a paper
cup repeatedly, it will land with the point up?
c. Combine your data set with those of other groups. What would you now say is
the probability that if a thumbtack is tossed from a paper cup repeatedly, it will
land with the point up?
d. Which probability estimate do you trust more? Why?
Activity 1 is an example of assigning a probability experimentally (or empirically—
that is, through experience). You performed a process a large number of times, and
then you used that information to formulate a probability statement. In the experiment,
you noted the position of the thumbtack each time. That position is an outcome. The
term outcome is used to mean one of the simplest results of an experiment. All of the
outcomes together form what is called a sample space so the sample space for this
experiment has just two possible outcomes, point up and point sideways. You may
have determined the probability of the event that a thumbtack lands with the point up
by seeing what fraction of all the tosses the thumbtack landed point up. We estimate
the probability of the event of a thumbtack landing point up by the fraction
the number of times a thumbtack landed point up
the total number of times the thumbtack was tossed
.
The terms outcome, sample space, and event are all important for understanding and
determining both experimental and theoretical probabilities.
A sample space is the set of all possible outcomes of an experiment. If A
represents an event, then P(A) represents the probability of event A. P(A) =
number of times A happens
if all the outcomes are equally likely. Sometimes
total number of outcomes in the sample space
the probability of an event is called the relative frequency of the event.
EXAMPLE 4
Suppose a polling company called 300 people, selected at random. All were asked if
they would vote for a new school bond. Of the 300 people interviewed, 123 said yes,
134 said no, and 43 were undecided. The sample space of all possible outcomes is
yes, no, and undecided. An event Y might be getting an answer of yes. P(Y) = 123
300 .
This fraction is also called the relative frequency of getting a yes response.
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Chapter 27
Notes
A diagram such as the one to the right (called a Venn
diagram) is occasionally useful in thinking about
sample spaces and their outcomes and events. You
imagine that all of the outcomes are inside the
sample-space box, with those favoring event A inside
the circle labeled “Event A.” Thus if the sample space
contains all the outcomes of tossing a thumbtack, A
could represent the event that the point is up.
Similarly, the experiment described in Example 4
might give a Venn diagram such as the one to the
right. In Venn diagrams, the sizes of the regions need
not reflect the numbers of votes.
Quantifying Uncertainty
Everything in the box is
in the sample space.
Not event A
Event A
Everything in the box is
in the sample space.
Yes
No
Undecided
THINK ABOUT . . .
What is the sample space for randomly choosing a letter of the alphabet?
For this experiment, make a Venn diagram that shows the event “chose a
vowel.” For this experiment it is easy to be explicit about the outcomes.
When tossing thumbtacks in Activity 1, it may have occurred to you that for efficiency’s sake, you could toss more than one thumbtack at a time and note how many
of them land point up and how many land point sideways. For example, your group
could have tossed 10 thumbtacks at a time and gathered data more quickly by viewing the single toss as 10 repetitions of the basic one-thumbtack experiment. This approach is certainly possible, but because it literally is a different experiment from
tossing just one thumbtack, it opens up other possibilities for descriptions of outcomes.
For a toss of 10 thumbtacks (with numbers 1 through 10 painted on for clarity), you
could view the outcomes differently from before. If you use S to mean that a thumbtack pointed sideways and U to mean it pointed up, a possible outcome could now be
denoted by SSSUSUUSUS. Other outcomes are possible of course, such as
USUUSUSSSU. The sample space would then consist of all possible lists with 10 Ss
and Us. From this setup, an example of an event such as “all the outcomes in which 7
thumbtacks point up” makes sense.
Sometimes it is difficult to distinguish between outcomes and events. An event consists of one or more outcomes that are of particular interest, such as the event of getting a head and a tail on the toss of a penny and a dime. This event consists of two
possible outcomes: a head on the penny and a tail on the dime, or a tail on the penny
and a head on the dime.
But what if we toss only the dime, and the event is getting heads? Now the only outcome that satisfies this condition is getting heads when tossing the dime, so the outcome and the event are the same: getting heads on a toss of a dime. A clear
Section 27.2
635
Methods of Assigning Probabilities
understanding of what the sample space is can help to clarify the distinction between
outcome and event.
Discussion 4
Some Facts about Probabilities
1. Why is a probability limited to numbers between, and including, 0 and 1? When
is it 0? When is it 1?
2. Suppose you calculate the probability of each possible outcome of an experiment, and add these probabilities. What should the sum be?
3. Suppose the probability of an event A is
2
3
. What is the probability of event A
not occurring?
Consider the sample space for drawing one ball from a bag containing 2 red balls and
3 blue balls, replacing the ball, and then drawing again. This sample space consists of
all possible outcomes and is made much easier by thinking of the balls as numbered,
that is, r1, r2, b1, b2, and b3. Counting the possible outcomes requires that we list all
possible pairings. Listing all outcomes should be done in a systematic, organized way
to assure that all pairs are included and none are counted twice. The following shows
a systematic listing:
r1r1
r1r2
r1b1
r1b2
r1b3
r2r1
r2r2
r2b1
r2b2
r2b3
b1r1
b1r2
b1b1
b1b2
b1b3
b2r1
b2r2
b2b1
b2b2
b2b3
b3r1
b3r2
b3b1
b3b2
b3b3
EXAMPLE 5
Suppose someone is interested in drawing two red balls. Drawing two red balls is
then the event of interest. The four possible outcomes in this event are r1r1, r1r2,
r2r1, and r2r2. The sample space has 25 possible outcomes in all. Thus the probability of drawing two reds is 45 .
THINK ABOUT . . .
What patterns do you notice in the listed sample space of 25 outcomes for
drawing two balls? If there had been 3 red balls and 3 blue balls, how
would the listing change? If there had been only 2 blue balls and 2 red
balls, how would the listing change?
It is even handy on occasion to think of an impossible event, such as getting two
greens in the last experiment. The probability of an impossible event is, of course,
equal to 0. Likewise, we can think of an event that is certain. For example, the event
of getting two balls that are either red or blue will happen every time. Thus, this
event is certain to happen and has probability 1.
Notes
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Chapter 27
Notes
Quantifying Uncertainty
THINK ABOUT . . .
What is the probability of NOT getting a red ball on the first draw and a
blue ball on the second draw? What is the probability of getting either red
or blue on the first draw? What is the probability of getting a red and then a
green?
Activity 2
Heads Up
1. Suppose you toss a coin a large number of times. Predict what fraction of those
times you would expect the coin to land heads up. Does your fraction give an experimental probability or a theoretical probability? Explain.
2. Try this experiment: Toss a coin ten times and note what is on top. How many
times did it land heads up? Combine your results with those from other people,
and compare your prediction in problem 1 about the probability of a coin landing
heads up with the experimental probability of a coin landing heads up.
In the first problem in Activity 2, you probably made the assumption that coins are
made in such a way that it is equally likely that a coin will land heads up as it is that
the coin will land tails up (or at least it is very close to equally likely). Using this assumption, we can formulate a probability statement without actually performing the
process a large number of times.
Recall that a theoretical probability is assigned based on determining the fraction of
times an event will occur under ideal circumstances. Thus, the theoretical probability
that a tossed fair coin will land heads up is 12 . You probably found in the second part
of Activity 3 that your experimental probability was close to the theoretical probability, but you may not have if you tossed the coin only ten times.
THINK ABOUT . . .
For some event, when both its experimental probability and its theoretical
probability are possible to determine, will the two probabilities be equal?
Explain.
Return to the list of outcomes in the sample space for the experiment in which two
balls were drawn (with replacement of the first ball) from a bag containing 2 red balls
and 3 blue balls. If the balls were identical in every other way then any one outcome
is as likely as any other outcome in the list of 25 outcomes. The outcomes are equally
likely. We can thus easily determine theoretical probabilities. Suppose we are interested in the event of drawing a red ball followed by a blue ball. This event occurs 6
out of 25 times in the sample space, so P(RB) = 256 .
Sometimes probabilities can be found either theoretically or experimentally. Other
times, such as when tossing a thumbtack (for which there are not equally likely outcomes), finding the probability experimentally is necessary.
Section 27.2
THINK ABOUT . . .
Suppose the balls in the bag differ in size and weight. Do you think the
outcomes are equally likely? If not, how can you find P(RB)?
Activity 3
Two Heads Up
1. If you toss a penny and a nickel, what is the probability of getting two heads?
How could you answer this question experimentally?
2. To determine the probabilities in Problem 1 theoretically it is helpful to list the
complete sample space.
Penny
H
H
T
T
Nickel
H
T
H
T
Is each of the four outcomes equally likely? What is the probability of getting two
heads? What is the probability of NOT getting two heads? How do you know?
The sample space in Activity 3 can give you information about other events. What is
the probability of getting one head and one tail? (Notice from the list that this can
happen in two ways.) What is the probability of getting no heads? Note that P(no
heads) = 1 – P(one or more heads). More generally, P(not A) = 1 – P(A), for any
event A.
Activity 4
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Methods of Assigning Probabilities
Give It a Spin
Spinners like those in board games can easily provide classroom experiments. Each
experiment works the same way: Flip the arrow on the spinner and note where the
arrow points. If the arrow lands on a line, spin again. (A partially-straightened paper
clip, with a pencil tip at one end, makes a serviceable spinner.)
red
red
white blue
white blue
A paper clip version
Suppose the spinner pictured above is used in an experiment of one spin.
1. What is the sample space for the experiment?
2. What angles are associated with the red, white, and blue outcomes?
Notes
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Chapter 27
Notes
Quantifying Uncertainty
3. What is the theoretical probability of each outcome?
4. What is P(not blue)?
5. How would you determine experimental probabilities for this experiment?
In sports, the term odds is often used. Statements about odds are not themselves
probability statements, but they can give probability statements as follows. If the
odds of Team A winning are given as “a to b,” this means that
P(A winning) =
a
a+b
and P(A losing) =
b
a+b
.
For example, if the odds in favor of Team A are “7 to 5,” then the anticipated P(A
winning) is equal to 127 and P(A losing) is equal to 125 . The statement, P(A winning)
is equal to
7
12
, means that if the game were played many, many times, you would
expect Team A to win
7
12
of the games. Odds can arise in other contexts. For exam-
ple, a doctor might say, “The odds of recovery are 3 to 2,” indicating that the probability of recovery is 3+3 2 = 53 .
THINK ABOUT . . .
What does “The odds of recovery are 3 to 2” mean, if the doctor has in
mind many, many cases?
T AKE-AWAY MESSAGE . . . Some vocabulary makes it easier to discuss a particular process
and what happens: sample space, outcomes, events, equally likely outcomes. It may be
possible to quantify the probability of an event in two ways: (1) by doing the experiment
many times to get an experimental probability, or (2) by reasoning about the situation to
get a theoretical probability. When an event is sure to happen, the probability of that event
is 1. When an event cannot occur, the probability of that event is 0. If one knows the
probability of a certain event A, then the probability of that event not occurring is P(not
A), and P(not A) = 1 – P(A). The sum of the probabilities of all outcomes of an experiment is 1. Statements about odds can be changed to probability statements. If the odds in
favor of event A are a to b, then the probability of A is a a+ b . ♦
Learning Exercises for Section 27.2
1. Explain how each situation is similar to tossing a coin.
a. Predicting the sex of an unborn child.
b. Guessing the answer to a true-false question.
c. Picking the winner of a two-team game.
d. Picking the winner of a two-person election.
2. Explain how each situation may NOT be similar to tossing a coin.
a. Predicting the sex of an unborn child.
Section 27.2
639
Methods of Assigning Probabilities
Notes
b. Guessing the answer to a true-false question.
c. Picking the winner of a two-team game.
d. Picking the winner of a two-person election.
3. Answer the question in each situation.
a. A medical journal reports that if certain symptoms are present, the probability of having a particular disease is 90%. What does that percentage mean?
b. A weather reporter says that the chances of rain are 80%. What does that that
percentage tell you?
c. Suppose the weather reporter says that the chances of rain are 50%. Does
that percentage mean that the reporter does not know whether it will rain
or not?
4. Is it possible for the probability of some event to be 0? Explain.
5. Is it possible for the probability of some event to be 1? Explain.
6. Is it possible for the probability of some event to be 1.5? Explain.
7. Three students are arguing. Chad says, “I think the probability is 1 out of 2.”
Tien says, “No, it is 40 over 80.” Falicia says, “It is 50%.” What do you say to
the students?
8. Suppose a couple is having difficulty choosing the name for their soon-to-beborn son. They agree that the first name should be Abraham, Benito, or Charles,
and the middle name should be Aidan or Benjamin, but they cannot decide which
to choose in either case. They decide to list all the possibilities and choose one
first-name, middle-name combination at random. Is this a probabilistic situation?
If so, what is the process that is being repeated?
9. Give the sample space for each of the following spinner experiments, and give
the theoretical probability for each outcome. (Assume the arrow originates at the
intersection of the lines within the figure.) How did you determine your theoretical probabilities? In this problem and others with drawings, assume that the angle
sizes are as they appear—90˚, 60˚, 45˚, and so on. Recall that the sum of all the
angles at the center of a circle must equal 360˚.
a.
b.
Q R
S
V
U
c.
W
Z
T
E
G F
Y
X
d.
H
J
I
10. How would you determine experimental probabilities for each outcome in the
spinner experiments in Learning Exercise 9? Explain your reasoning.
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Quantifying Uncertainty
11. Using the spinner in Learning Exercise 9(d) 12 times, is it possible to get J 8
times? Explain.
12. Consider the spinner shown to the right in
which the angle forming sector X is 120˚. What
is the probability of . . .
a. getting X?
b. not getting X?
c. getting Y or X?
d. getting Y and Z (simultaneously)?
e. getting Z?
f. not getting Z?
X
Y
Z
13. a. If P(E) is the probability of an event happening, we know that the probability
of the event not happening is 1 – P(E).
Explain why this is so.
b. What is the probability of not drawing an ace from a deck of cards?
c. What is the probability of a thumbtack not landing with the point up when
tossed, based on your results from Activity 1?
d. What is the probability of not getting two heads on the toss of a penny and a
nickel? (Be careful with this one. Hint: What is the sample space?)
e. What is the probability of not getting a 5 on the toss of one die?
f. You have a key ring with five keys, one of which is the key to your car.
What is the probability that if you blindly choose one key off the ring, it will
not be the key to your car? What assumption(s) are you making?
14. One circular spinner is marked into four regions. Region A has an angle of 100˚
at the center of the circle, Region B has an angle of 20˚ at the center, and Regions C and D have equal angles at the center.
D
A
B
C
a. Give the probability of each of the four outcomes with this spinner.
b. Are any of the outcomes equally likely?
c. What is the probability of not getting B?
15. Make sketches that describe the following spinners. Give the angle sizes of the
regions.
a. A spinner with 5 equally likely outcomes. (How could you do this accurately?)
b. A spinner with 5 outcomes of which 4 outcomes are equally likely. The fifth
outcome has probability twice that of each of the others.
Section 27.2
641
Methods of Assigning Probabilities
c. A spinner with 5 outcomes of which 3 outcomes have the same probabilities
and the fourth and fifth outcomes each have the same probability that is three
times the probability of each of the first three outcomes.
d. Two spinners that could be used to practice the basic multiplication facts
from 0 × 0 through 9 × 9. (Is each fact equally likely to be practiced with
your spinners?)
e. Two spinners that could be used to practice the “bigger” basic multiplication
facts, from 5 × 5 through 9 × 9.
16. List some situations in which the probability of a certain-to-happen event is 1.
List some situations in which the probability of a certain-not-to-happen event
is 0.
17. For each of the following situations, decide whether the situation calls for determining a probability experimentally or theoretically. If the probability is obtained
experimentally, describe how you would determine it, and if the probability is
theoretical, describe how you would find it.
a. Drawing a red M&M from a package of M&Ms.
b. Getting a two on the throw of one die.
c. Selecting a student who is from out of state on a particular college campus.
d. Selecting a heart from a regular deck of cards.
18. Consider the spinner to the right.
a. How many times would you expect to get S in
S
1200 spins?
b. How many times would you expect to get T in
T
1200 spins?
c. Is it possible to get S 72 times in 100 spins?
Explain.
d. Is it possible to get T 24% of the time in 250 spins? Explain.
e. What would you expect to get in 200 spins?
f. You likely used proportional reasoning in parts (a)-(e). How is proportional
reasoning being used?
19. In an experiment you are to draw one ball from a container (without looking, of
course) and note the ball’s color. You win if you draw a red ball. Each part below
has a description of the contents of two containers. For each part, which container provides the better chance of winning? If the chances are the same, say so.
Explain your reasoning.
Container 1
Container 2
a. 2 reds and 3 blues
or
6 reds and 7 blues
b. 3 reds and 5 blues
or
27 reds and 45 blues
c. 3 reds and 5 blues
or
16 reds and 27 blues
d. 3 reds and 5 blues
or
623 reds and 623 blues
20. What conditions (for example, for the container, for the balls, how the drawing
is done, and so on) are necessary to make reasonable the assumption that the
outcomes of a draw-a-ball experiment are equally likely?
Notes
642
Chapter 27
Notes
Quantifying Uncertainty
21. a. What probability statement says the same thing as “The odds for the
Mammoths winning are 3 to 10”?
b. What probability statement says the same thing as “The odds for the Dinosaurs losing are 6.7 to 1”?
22. a. If Team A is actually expected to win over Team B, how are x and y related
in the statement, “The odds in favor of Team A losing are x to y”?
b. If Team A and Team B play many, many times, what does the odds statement in part (a) mean?
23. Rephrase each of the following in terms of odds.
a. The probability of making the field goal is
2
3
.
b. The probability of drawing a red ball from the bag is
c. The probability of not getting a winning card is
7
30
7
12
.
.
27.3 Simulating Probabilistic Situations
In the previous section we discussed ways of finding probabilities experimentally.
We now undertake doing an experiment a large number of times and noting the
outcomes.
Activity 5
How Many Heads?
Suppose you want to find the probability of obtaining three heads and a tail when
four coins are tossed. You could, of course, toss four coins a large number of times,
and count the number of times that you obtain exactly three heads. The probability
would be as follows:
the number of times exactly three of the four coins show heads
the total number of times four coins were tossed
.
In your group, toss four coins 10 times and record the number of times you got 3
heads. Combine your results with those of other groups to find the probability of obtaining 3 heads on a toss of three coins.
Determining experimental probabilities can be very time consuming when one
actually carries out an experiment many, many times. Because numbers selected randomly are equally likely, we can use them to speed up this process. Using randomly
selected numbers to find a probability in this manner is called a simulation.
Section 27.3
643
Simulating Probabilistic Situations
Notes
A randomly selected number or object from a set of numbers or objects is one
that has a chance of being selected that is equal to that of any other number or
object in the set.
Activity 6
”Tossing” Four Coins Again
Once again, we will experimentally find the probability of obtaining three heads on a
toss of four coins. This time we will simulate tossing four coins 100 times. To do so,
compile a set of 100 four-digit numbers. There are many ways to do this. Your
instructor will tell you whether to use the TI-73, Fathom, Excel, Illuminations, or a
Table of Randomly Selected Digits (TRSD). Each of these methods is explained in
an appendix. See Appendix H (using the TI-73), I (using Fathom), J (using Excel), K
(using the Illuminations website), or L (using a TRSD) and following the given
directions.
Once you have the set of 100 four-digit numbers, consider the digits in each number.
An even digit can be used to represent heads, and an odd digit to represent tails.
How many of the 100 four-digit numbers have three even digits and one odd digit?
This number, call it x, can be used to find the probability of tossing three heads:
x
. How does this probability match the probability you found when
P(3H) = 100
actually tossing coins in Activity 5?
There are many types of probabilistic situations in which a probability can be found
with a simulation using a set of randomly selected numbers. For example, one can
simulate drawing colored balls from a bag. Suppose we have a paper bag with two
red balls and three blue balls. We want to know the probability of drawing a red
ball on the first draw, replacing it, and then drawing a blue ball on the second draw.
Keep in mind that the first ball is replaced, that is, we are drawing balls with
replacement.
Because there are five balls, a set of random numbers 1–5 can be used to simulate
drawing with replacement. For example, 1 and 2 can represent red balls and 3, 4, and
5 can represent blue balls.
THINK ABOUT . . .
How can this method also tell you the number of times you draw a blue
ball followed by another blue ball? Why is it important to know that the
first ball was replaced? (Hint: Think of outcomes such as 33, 44, 55)
Activity 7
Red and Blue
Return to the method you used in Activity 6 to generate random numbers from 1 to 5.
Make a table with two columns, one for red and the other for blue. Each will have
randomly generated numbers from 1 to 5. Let 1 and 2 represent red balls and 3, 4,
and 5 represent blue balls. (In the table, having a 2 in the first column and a 4 in the
second column would represent drawing a red, replacing it, and then drawing a blue.)
644
Chapter 27
Notes
Quantifying Uncertainty
1. Do 20 trials, and find the experimental probability of drawing a red ball and then
a blue ball.
2. Combine your 20 trials with those of others, and from the combined data find the
experimental probability of drawing a red and then a blue.
3. Use the same pairs of numbers to determine the probability of drawing two blues,
first using just your data and then using the combined data from everyone.
In Problem 1 of Activity 7, would you have a different result if you did 100 trials
instead of 20 trials? Which would you trust to be the better estimate of the probability? One conclusion you may have reached by now is that a large number of trials of
a simulation will lead to a more accurate estimate of a probability.
Just as we used random numbers to simulate the drawing of balls from a bag, the
drawing of the balls itself can be a simulation of some other event. What matters is
that the objects (coins, balls, or other objects) are used to represent something else.
For example, the toss of one coin may be used to simulate the unknown gender of a
baby. Whether using some type of simulation or actually listing all elements in the
sample space (which may not always be possible), a simulation is useful when it
models some actual situation. For example, consider a lake having only bass (40%)
and catfish (60%). Catching a bass and then a catfish can be simulated with a bag of
100 balls, 40 of which are one color and 60 of which are another color. Or you could
consider 100 random two-digit numbers (00 through 99), where 00 through 39 could
represent bass and 40 through 99 could represent catfish.
THINK ABOUT . . .
Could the bass and catfish draws be represented using just 5 balls? How?
Using one-digit numbers 0 through 9? How?
Activity 8
Let’s Spin
Explain how you could use the following spinner to simulate the drawing, with replacement, of two balls from a bag in which two balls are red, three are blue, and the
balls are otherwise identical. (Assume all five sectors have the same central angle.)
B
R
B
R
B
Many discussions of probability focus on experiments about tossing coins, throwing
dice, or drawing balls from containers. At first it may seem silly to spend much time
on such experiments, especially if they have to do only with games. But actually
Section 27.3
these activities can be used to model a variety of real-life probabilistic settings. For
example, suppose you are a doctor with five patients who have the same bad disease.
You have an expensive treatment that is known to cure the disease with probability
1
4 , or 25%. You are wondering what the probability of curing none of your five
patients is, if you use the expensive treatment. Perhaps surprisingly, this situation can
be simulated by drawing cards from a hat! Draw from a hat that has four identical
cards except for the markings on the cards. One card is marked cured and 3 cards are
marked with not cured. Drawing these cards in repeated trials provides a model for
the 25% cure rate.
To model the situation with the five patients, you would draw a card, note what it
says, put it back in the hat, and then repeat this process four more times. After the
fifth draw is finished, you would see whether any cured cards had been drawn—that
is, whether any “patients” were cured. Because probabilities should be interpreted
over the long run, you would repeat this five-draws experiment many times to get an
idea of the probability that you are concerned about—the probability that none of
your patients would be cured.
THINK ABOUT . . .
Why would you need to replace the card in this situation of the doctor with
five patients?
You could simulate the same situation in many ways—for example, with one green
ball and three red balls in a paper bag. Drawing a green ball would represent a cure,
and drawing a red ball would represent no cure.
Discussion 5
Representing Situations as Drawing Balls from a Bag
or Drawing Numbers from a Hat
Explain how each situation could be modeled by an experiment in which balls are
drawn from a bag. Tell what each color would represent, how many colors of balls
would be involved, and how you would proceed. How could each situation be
modeled by drawing numbers from a hat?
a. Predicting the sex of a child.
b. Deciding on the chances of four consecutive successful space shuttle launches, if
the probability of success each time is 99%.
c. Getting a six on a toss of an honest die.
d. A beginner who is 10% accurate shooting an arrow at a balloon and hitting it.
e. An expert who is 99% accurate shooting an arrow at a balloon and hitting it.
f.
645
Simulating Probabilistic Situations
Predicting whether a germ will survive if it is treated with chemical X that kills
75% of the germs.
g. The chances that 10 pieces of data sent back from space are all correct if the
probability of each piece being correct is 0.9.
Notes
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Chapter 27
Notes
Quantifying Uncertainty
The processes in Discussion 5 depend on random draws of balls (or numbers, in the
case of drawing a number from a hat). Hence, each ball has an equal chance of being
drawn. For example, consider again drawing a ball from a bag of 2 red balls and 3
blue balls, replacing it, and then drawing again. The sample space for this experiment is made up of these equally-likely outcomes:
r1r1
r1r2
r1b1
r1b2
r1b3
r2r1
r2r2
r2b1
r2b2
r2b3
b1r1
b1r2
b1b1
b1b2
b1b3
b2r1
b2r2
b2b1
b2b2
b2b3
b3r1
b3r2
b3b1
b3b2
b3b3
Recall that when outcomes are equally likely, it is possible to find the theoretical
probability of an event by counting the outcomes associated with the event, and the
number of outcomes for an event
probability of the event is then number
of outcomes in the sample space . But what happens
if the balls are not replaced? We next discuss drawing balls without replacement.
Discussion 6
Drawing Two Balls Without Replacement
We now change the situation so that we draw two balls from a bag containing 2 red
balls and 3 blue balls, but we do not replace the first ball before drawing the second
ball.
1. Write the sample space for this without-replacement situation. How is it different
from the 25-outcome sample space above? (Hint: If a ball is not replaced, what
balls are available for the next draw?)
2. What are some possible outcomes for this experiment?
3. What are some events that could be considered?
THINK ABOUT . . .
From Discussion 6, what patterns do you notice in your list of outcomes
when drawing two balls without replacement? If there had been 3 red balls,
how would the listing change? If there had been only 2 blue balls, how
would the listing change?
Knowledge of probability is useful in understanding many types of situations. The
problem in Activity 9 is just one example.
Activity 9
Free Throws
Tabatha is good at making free throws, and in the past she averaged making 2 out of
every 3 free throws. At one game, she shot 5 free throws, and she missed every one!
Her fans insist she must have been sick or hurt, or that something must have been
wrong. They say it was not possible for her to miss all 5 shots. Are they correct?
Is Tabatha’s missing 5 three throws impossible? That is, is the probability of this
event 0?
Section 27.3
Here is a way to simulate this situation: Suppose we consider two-digit random numbers, from 01 to 99 (ignore 00). If the numbers are random, then 23 of the time they
should be numbers from 01 to 66, and
1
3
of the time they should be numbers from 67
to 99. (Could we use 00–65 and 66–98, ignoring 99?) Randomly find five pairs of
two-digit numbers. Let all numbers from 01 through 66 represent free throws made
and all numbers from 67 through 99 as free throws missed. Work in your groups to
obtain 20 sets of five throws, combine them with others, and predict the probability
of this event.
T AKE-AWAY MESSAGE . . . It may be surprising that, with clever choices of coding the numbers, a set of randomly selected numbers can be used to simulate so many situations. Different repetitions of the same simulation illustrate that different repetitions will most often
give different, but close, experimental probabilities. In addition to random numbers, other
entities such as balls drawn from a bag can be used to simulate a wide variety of events.
In all cases, listing all the possible outcomes gives you the sample space and makes
defining events and finding probabilities easier. ♦
Learning Exercises for Section 27.3
1. Set up and complete a simulation of tossing a die 100 times. How do the probabilities of each outcome compare with the theoretical probabilities? Do the same
simulation for 500 repetitions. What pattern do you notice as the number of repetitions increases?
2. a. Set up and complete a simulation to find the probability of getting green
twice in a row if a spinner is on a circular region that is 13 green, 16 blue,
1
3
b.
c.
d.
e.
647
Simulating Probabilistic Situations
red, and the rest yellow. Carry out the simulation 30 times and record the
outcomes. (Your record should include colors.) Answer parts (b) through (e)
based on the outcomes of your experiment, and tell whether your experimental values are close to the theoretical values.
Which outcome is most likely? Why?
What is the probability of getting a green on the first spin and a blue on the
second?
What is the probability of getting a green on one of the spins and a blue on
the other? Why is this question different from the question in part (c)?
What is the probability of not getting green twice in a row? (Hint: There is
an easy way.)
3. Earlier in this section the following situation was simulated using cards: Suppose
you are a doctor with five patients who have the same bad disease. You have an
expensive treatment that is known to cure the disease with a probability of 25%.
You are wondering what the probability of curing none of your five patients is, if
you use the expensive treatment. Set up and complete a different simulation of
this situation. (Because the question is about a group of five patients, use groups
of size 5.) You decide how many times to do the simulation.
Notes
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Chapter 27
Notes
Quantifying Uncertainty
4. On one run of the free-throw simulation in Activity 9, the event YYYYN had a
proportion 0.319. What does that mean?
5. You have 4 markers in a box. One is labeled N, one is labeled O, one is labeled
D, and one is labeled E. Set up and complete a simulation to find the probability
that if the labels are drawn, without replacement, one at a time from the box, they
will spell DONE.
6. Does a simulation using randomly generated numbers give theoretical or experimental probabilities? Explain.
7. a. Go to http://illuminations.nctm.org/, and click on Activities. In the options,
type in Adjustable Spinner. Set the probabilities for the four colors in any
way you want by moving the dots on the circle OR by moving the buttons for
each color OR by setting the percents of each circle in the results frame. Set
Number of spins to 1, and click on Spin. You will see how the spinner
works. Write down the numbers in the results frame. These are the relative
sizes of the regions of the circles and show theoretical probabilities (but in
percents). You can open five colors by adding a sector (click on +1). Once
again set the probabilities. Open the screen to show the table at the bottom.
b. Use this spinner activity to simulate a 5-outcome experiment with unequally
likely outcomes, and run the simulation 100 times. How do the experimental
results compare with the theoretical ones in the table at the bottom of the
screen?
c. Repeat with a run of 1000 simulations. How do the experimental results
compare with the theoretical ones?
d. Repeat with a run of 10,000 simulations. How do the experimental results
compare with the theoretical ones?
27.4 Issues for Learning: What Do Large-Scale Tests
Tell Us About Probability Learning?
Children develop some basic notions of chance as they interact with the world. They
begin to evaluate the likelihood of certain events, such as whether it will rain, and
think about such events as impossible, unlikely, likely, or certain. Yet most of their
thinking about probability is intuitive rather than based on evidence. In fact, it has
long been known that even adults’ ideas about probability (or chance or likelihood)
are not reliable. Intuitions are often not correct. For example, when tossing a coin
repeatedly, to think that a tails up is almost certain to occur after five consecutive
heads up is common, not only with children, but also with adults (thus the label, the
gambler’s fallacy). And adults certainly know that a coin does not have a memory!
Thus it is not surprising to see these same intuitive ideas in children.
In recognition of the increasing importance of probability (and statistics) in an
educated person’s life and the poor performance of adults in situations where probability plays a role, there has been more attention paid to probability (and statistics)
Section 27.4
Issues for Learning: What Do Large-Scale Tests Tell Us About Probability Learning?
in required schooling over the last several years, thus offering the important opportunity to learn (provided that the teachers do not skip the material). As a result,
the performance of U.S. students has exceeded the international average on some
i
probability items in international testing programs. For example, for the test items
given below, the U.S. average at eighth-grade was 62% on the first item and 77%
on the second item, with the respective international averages being 57% and 48%.
Having an opportunity to learn certainly makes a difference.
Item 1. If a fair coin is tossed, the probability that it will land heads up is 12 . In four
successive tosses, a fair coin lands heads up each time. What is likely to happen
when the coin is tossed a fifth time?
A. It is more likely to land tails up than heads up.
B. It is more likely to land heads up than tails up.
C. It is equally like to land heads up or tails up.
D. More information is needed to answer the question.
Item 2. The eleven chips shown below are placed in a bag and mixed.
2
10
3
5
12
11
6
14
8
18
20
Chelsea draws one chip from the bag without looking. What is the probability that
Chelsea draws a chip with a number that is a multiple of three?
A.
1
11
B.
1
3
C.
4
11
D.
4
7
THINK ABOUT . . .
The answer to both Items 1 and 2 is C. How do your answers match up
with the correct answers?
Testing programs do, however, suggest that the opportunity to learn does not extend
far enough. For example, only 28% of the twelfth graders were at least partially suci
cessful on the following more complicated test item. Again, more experience with a
complete description of all the outcomes might provide better performance.
Notes
649
650
Chapter 27
Quantifying Uncertainty
Notes
Item 3. The two spinners shown above are part of a carnival game. A player wins a
prize only when both arrows land on black after each spinner has been spun once. James
thinks that he has a 50-50 chance of winning. Do you agree? Justify your answer.
In answering Item 3, one student wrote, “No. He only has a
must multiply the two
1
2
1
4
chance because you
chances from each individual spinner.” Another wrote,
“No. They start at the same place but it depends on how hard or light each spinner is
spun.” Both were correct in deciding “No” but the reasons given are very different.
In this case, the first student provided a correct answer but not the second student.
Asking students to explain their reasoning is much more common nowadays that it
once was.
Students often do not appreciate the importance of knowing whether the outcomes
are equally likely. They are willing to assign equal probabilities to any experiment
that has, say, 3 outcomes. Spinners with unequal regions give one means of exposing
the children to the fact that outcomes may not be equally likely. Experiments like
tossing thumbtacks and noting whether the tip is up or down, or tossing a styrofoam
cup and recording whether the cup ends up on its side with its wide end down or
right side up, can also expose the students to outcomes that are not equally likely.
The notion that probabilities are long-run results rather than next-case results may
ii
not be clear to some students. Some of this problem may be due to not attending to
the fact that a probabilistic situation involves a process happening a large number of
times. For example, we may say, “What is the probability that we get red if we spin
the arrow on this spinner?” Although we understand this question to be about a fraction in the long run with many repetitions, students may interpret the question literally and focus on just the next toss. This thinking may give correct answers for some
experiments, but it may lead to an incorrect understanding of probability. Doing an
experiment many times when probability is first introduced may help to establish the
long-run nature of probability statements, with frequent reminders later on.
A narrow next-case focus may explain the following test results with fourth graders
i
in a national testing program. Even though 66% of the fourth graders were successful in choosing “1 out of 4” for Item 4 below, only 31% were successful in choosing
the answer “3 out of 5” for Item 5, where the next-case view showed three possibilities. In general, a question in a “1 out of n” situation is much easier than one about a
“several out of n” situation.
Section 27.5
651
Check Yourself
Item 4. The balls in this picture are placed in a box and a child picks one without
looking. What is the probability that the ball picked will be the one with dots?
A. 1 out of 4
B. 1 out of 3
C. 1 out of 2
D. 3 out of 4
Item 5. There are 3 fifth graders and 2 sixth graders on the swim team. Everyone’s
name is put in a hat and the captain is chosen by picking one name. What are the
chances that the captain will be a fifth grader?
A. 1 out of 5
B. 1 out of 3
C. 3 out of 5
D. 2 out of 3
THINK ABOUT . . .
What are some other possible reasons that performance on Item 5 was
worse than that on Item 4? How would you redesign the two items to make
them more comparable in terms of difficulty?
T AKE-AWAY MESSAGE . . . Children begin to form intuitive ideas about probability as they
mature, but many of these ideas are incorrect, and persist into adulthood. International
and national tests provide valuable information about what children know and don’t know
about probability. ♦
27.5 Check Yourself
Along with an ability to deal with exercises like those assigned and with experiences
such as those during class, you should be able to meet the following objectives.
1. Explain what is meant by the term random.
2. State and recognize the key features of a probabilistic situation, and rephrase a
given wording of a non-probabilistic situation to make it probabilistic.
3. Tell what the word “probability” means in the statement “the probability of that
happening is about 30%.” Express and interpret probability using the P(A)
notation.
4. Distinguish between experimental and theoretical probabilities, and describe how
either one or both could be used to determine a probability in a given situation.
5. For a given task, use the following vocabulary words accurately: outcome, sample space, and event.
Notes
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Chapter 27
Notes
Quantifying Uncertainty
6. Use the following important results:
•
If event A is certain to happen, P(A) = 1.
•
If event A is impossible, P(A) = 0.
•
For any event A, 0 ≤ P(A) ≤ 1, and P(A) = 1 – P(A).
7. Appreciate the usefulness of coin tossing, spinners, or drawing balls from a bag,
and design an experiment to simulate some situation using one or more of those
methods.
8. Find theoretical or experimental probabilities for a given event.
9. Design and carry out an experiment to simulate a situation using a TRSD (or
Excel, Fathom, the TI 73, or Illuminations). Show how you determined probabilities with the simulation.
10. Distinguish between drawing with replacement and drawing without replacement.
REFERENCES FOR CHAPTER 27
i
See http://isc.bc.edu and http://nces.ed.gov/nationsreportcard/ for test reports and, often, test items that have
been released to the public.
ii
Zawojewski, J. S., & Shaughnessy, J. M. (2000). Data and chance. In E. A. Silver & P. A. Kenney (Eds.),
Results from the seventh mathematics assessment of the National Assessment of Educational Progress,
pp. 235–268. Reston, VA: National Council of Teachers of Mathematics.