Algebra 2 Interactive Chalkboard
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Lesson 12-1
The Counting Principle
Lesson 12-2
Permutations and Combinations
Lesson 12-3
Probability
Lesson 12-4
Multiplying Probabilities
Lesson 12-5
Adding Probabilities
Lesson 12-6
Statistical Measures
Lesson 12-7
The Normal Distribution
Lesson 12-8
Binomial Experiments
Lesson 12-9
Sampling and Error
Example 1 Independent Events
Example 2 Fundamental Counting Principle
Example 3 More than Two Independent Events
Example 4 Dependent Events
A sandwich menu offers customers a choice of white,
wheat, or rye bread with one spread chosen from
butter, mustard, or mayonnaise. How many different
combinations of bread and spread are there?
First, note that the choice of the type of bread does not
affect the choice of the type of spread, so these events
are independent.
Method 1
Tree Diagram
Let W represent white, H, wheat, R, rye, B, butter, M,
mustard, and Y, mayonnaise. Make a tree diagram in
which the first row shows the choice of bread and the
second row shows the choice of spread.
Bread
Spread
Possible
Combinations
Answer: There are nine possible outcomes.
Method 2
Make a Table
Make a table in which each row represents a type of
bread and each column represents a type of spread.
Bread
White
Butter
WB
Spread
Mustard
WM
Mayo
WY
Wheat
Rye
HB
RB
HM
RM
HY
RY
Answer: This method also shows that there are
nine outcomes.
A pizza place offers customers a choice of American,
mozzarella, Swiss, feta, or provolone cheese with one
topping chosen from pepperoni, mushrooms or
sausage. How many different combinations of cheese
and toppings are there?
Answer: 15
Multiple-Choice Test Item
For their vacation, the Murray family is choosing a trip
to the beach or to the mountains. They can select their
transportation from a car, plane, or train. How many
different ways can they select a destination followed
by a means of transportation?
A 2
B 5
C 6
D 9
Read the Test Item
Their choice of destination does not affect their choice
of transportation, so these events are independent.
Solve the Test Item
There are two ways they can choose a destination and
3 ways they can choose a means of transportation. By
the Fundamental Counting Principle, there are 2 • 3 or
6 total ways to choose.
Answer: C
Multiple-Choice Test Item
For their vacation, the Esper family is going on a trip.
They can select their transportation from a car, plane,
or train. They can also select from 4 different hotels.
How many different ways can they select a
means of transportation followed by a hotel?
A 8
B 12
Answer: B
C 16
D 7
Communication How many codes are possible if an
answering machine requires a 2-digit code to
retrieve messages?
The choice of any digit does not affect the other digit, so
the choices of digits are independent events.
There are 10 possible choices for the first digit and 10
possible choices for the second digit.
Answer: So, there are
different codes.
or 100 possible
Banking Many automated teller machines (ATM)
require a 4-digit code to access an account. How
many codes are possible?
Answer: 10,000
How many different schedules could a student have
who is planning to take 4 different classes? Assume
each class is offered each period.
When a student schedules a given class for a given
period, he or she cannot schedule that class for any other
period. Therefore, the choices of which class to schedule
each period are dependent events.
There are 4 classes that can be taken during the first
period. That leaves 3 classes for the second period, 2
classes for the third period, and so on.
Period
Number of Choices
1st 2nd 3rd
4
3
2
4th
1
Answer: There are
or 24 different
schedules for a student who is taking
4 classes.
How many different schedules could a student have
who is planning to take 5 different classes?
Answer: 120
Example 1 Permutation
Example 2 Permutation with Repetition
Example 3 Combination
Example 4 Multiple Events
Eight people enter the Best Pic contest. How many
ways can blue, red, and green ribbons be awarded?
Since each winner will receive a different ribbon, order is
important. You must find the number of permutations of
8 things taken 3 at a time.
Permutation formula
Simplify.
1
1
1
1
1
1
1
1
1
1
Divide by common factors.
Answer: The ribbons can be awarded in 336 ways.
Ten people are competing in a swim race where 4
ribbons will be given. How many ways can blue,
red, green, and yellow ribbons be awarded?
Answer: 5040
How many different ways can the letters of the word
BANANA be arranged?
The second, fourth, and sixth letters are each A.
The third and fifth letters are each N.
You need to find the number of permutations of 6 letters
of which 3 of one letter and 2 of another letter are
the same.
Answer: There are 60 ways to arrange the letters.
How many different ways can the letters of the word
ALGEBRA be arranged?
Answer: 2520
Five cousins at a family reunion decide that three of
them will go to pick up a pizza. How many ways can
they choose three people to go?
Since the order they choose is not important, you must
find the number of combinations of 5 cousins taken three
at a time.
Combination formula
Simplify.
Answer: There are 10 ways to choose three people
from the five cousins.
Six friends at a party decide that three of them will go
to pick up a movie. How many ways can they choose
three people to go?
Answer: 20 ways
Six cards are drawn from a standard deck of cards.
How many hands consist of two hearts and
four spades?
By the Fundamental Counting Principle, you can multiply
the number of ways to select two hearts and the number
of ways to select four spades.
Only the cards in the hand matter, not the order in which
they were drawn, so use combinations.
C(13, 2)
C(13, 4)
Two of 13 hearts are to be drawn.
Four of 13 spades are to be drawn.
Combination formula
Subtract.
Simplify.
Answer: There are 55,770 hands consisting of 2 hearts
and 4 spades.
Thirteen cards are drawn from a standard deck of
cards. How many hands consist of six hearts and
seven diamonds?
Answer: 2,944,656 hands
Example 1 Probability
Example 2 Probability with Combinations
Example 3 Odds
Example 4 Probability Distribution
When three coins are tossed, what is the probability
that all three are heads?
You can use a tree diagram to find the sample space.
First Coin
Second Coin
Third Coin
Possible
Outcomes
There are 8 possible outcomes. You can confirm this
using the Fundamental Counting Principle. There are 2
possible results for the first coin and 2 for the second coin,
so there are
or 8 outcomes. Only one of these
outcomes is HHH so
The other outcomes are
failures, so
Probability formula
Answer: The probability of tossing three heads
is
When three coins are tossed, what is the probability
that exactly two are heads?
Answer:
Roman has a collection of 26 books–16 are fiction and
10 are nonfiction. He randomly chooses 8 books to
take with him on vacation. What is the probability that
he chooses 4 fiction and 4 nonfiction?
Step 1
Determine how many 8-book selections
meet the conditions.
C(16, 4)
Select 4 fiction books.
Their order does not
matter.
C(10, 4)
Select 4 nonfiction.
Step 2
Use the Fundamental Counting Principle to
find the number of successes.
Step 3
Find the total number, s + f, of possible
8-book selections.
Step 4
Determine the probability.
P(4 fiction, 4 nonfiction)
Probability formula
Substitute.
Use a calculator.
Answer: The probability is about 0.24464
or 24.5%.
Ainsley has a collection of 15 CDs–5 are jazz and 10
are blues. She randomly chooses 7 CDs to take with
her on vacation. What is the probability that she
chooses 2 jazz and 5 blues?
Answer: The probability is about 0.392
or 39.2%.
Life Expectancy According to the U.S. National Center
for Health Statistics, the chances of a male born in
1990 living to be at least 65 years of age are about
3 in 4. For females, the chances are about 17 in 20.
What are the odds that a male born in 1990 will die
before age 65?
Three out of four males will live to be at least 65, so the
number of successes (living to 65) is 3. The number of
failures is
Odds formula
Answer: The odds of a male dying before age 65
are 1:3.
What are the odds that a female born in 1990 will die
before age 65?
Seventeen out of twenty females will live to be at least
65, so the number of successes in this case is 17. The
number of failures is
Odds formula
Answer: The odds of a female dying before age 65
are 3:17.
Life Expectancy The chances of a male born in 1980
to live to be at least 65 years of age are about 7 in 10.
For females, the chances are about 21 in 25.
a. What are the odds that a male born in 1980 will live to
age 65?
Answer: 7:3
b. What are the odds that a female born in 1980 will live
to age 65?
Answer: 21:4
Suppose two dice
are rolled. The
table and the
relative-frequency
histogram show
the distribution of
the sum of the
numbers rolled.
S = Sum
Probability
15
1
12
18
36
9
6
2
3
4
5
6
7
8
9
10
11
12
Use the graph to
determine which
outcomes are
least likely. What
are their
probabilities?
Answer: The least probability is
The least
likely outcomes are sums of 2 and 12.
S = Sum
15
1
12
18
36
9
6
2
3
4
5
6
7
8
9
10
11
12
Probability
Use the table to find
What other sum with
this probability is a sum of 3?
Answer:
probability is a sum of 3.
The other sum with this
What are the odds of rolling a sum of 5?
Step 1
Step 2
Identify s and f.
Find the odds.
Answer: So, the odds of rolling a sum of 5 are 1:8.
Suppose two dice are
rolled. The table and
the relative frequency
histogram show the
distribution of the
sum of the numbers
rolled.
S = Sum
Probability
15
1
12
18
36
9
6
2
3
4
5
6
7
8
9
10
11
12
a. Use the graph to
determine which
outcomes are the
second most likely.
What are their
probabilities?
Answer:
S = Sum
15
1
12
18
36
9
6
2
3
4
5
6
7
8
9
10
11
Probability
b. Use the table to find
has the same probability?
Answer:
What other sum
12
S = Sum
15
1
12
18
36
9
6
2
3
4
5
6
7
8
9
Probability
c. What are the odds of rolling a sum of 3?
Answer: 1:18
10
11
12
Example 1 Two Independent Events
Example 2 Three Independent Events
Example 3 Two Dependent Events
Example 4 Three Dependent Events
Gernardo has 9 dimes and 7 pennies in his pocket. He
randomly selects one coin, looks at it, and replaces it.
He then randomly selects another coin. What is the
probability that both of the coins he selects
are dimes?
Explore
These two events are independent since he
replaces the coin. The second event is not
affected by the first.
Plan
Since there are 16 coins, the probability of
getting a dime is
Solve
Probability of
independent events
Substitute
and multiply.
Answer: The probability that both coins are dimes
is
or about 31.6%.
Examine
You can verify this by making a tree diagram
that includes probabilities. Let D stand for
dimes and P stand for pennies.
Gernardo has 9 dimes and 7 pennies in his pocket. He
randomly selects one coin, looks at it, and replaces it.
He then randomly selects another coin. What is the
probability that both of the coins he selects
are pennies?
Answer:
or about 0.19
When three dice are rolled, what is the probability
that two dice show a 5 and the third die shows an
even number?
Let A be the event that the first die is a 5.
Let B be the event that the second die is a 5.
Let C be the event that the third die is even.
Probability of
independent events
Substitute and multiply.
Answer: The probability that two dice show a five
and the third die shows an even number
is
When three dice are rolled, what is the probability
that one die is a multiple of 3, one die shows an
even number, and one die shows a 5?
Answer:
The host of a game show draws chips from a bag to
determine the prizes for which contestants will play.
Of the 20 chips in the bag, 11 show computer, 8 show
trip, and 1 shows truck. If the host draws the chips at
random and does not replace them, find
each probability.
a computer, then a truck
Dependent events
After the first chip is
drawn, there are 19 left.
Answer: The probability of a computer then a trip
is
or about 0.03.
two trips
Dependent events
If the first chip shows
trip, then 7 of the
remaining 19 show trip.
Answer: The probability of the host drawing two trips
is
The host of a game show draws chips from a bag to
determine the prizes for which contestants will play.
Of the 20 chips in the bag, 11 show computer, 8 show
trip, and 1 shows truck. If the host draws the chips at
random and does not replace them, find
each probability.
a. a truck, then a trip
Answer:
b. two computers
Answer:
Three cards are drawn from a standard deck of cards
without replacement. Find the probability of drawing a
heart, another heart, and a spade in that order.
If the first two cards are
hearts, then 13 of the
remaining cards
are spades.
Answer: The probability is
or about 0.015.
Three cards are drawn from a standard deck of cards
without replacement. Find the probability of drawing a
diamond, another diamond, and another diamond in
that order.
Answer:
Example 1 Two Mutually Exclusive Events
Example 2 Three Mutually Exclusive Events
Example 3 Inclusive Events
Sylvia has a stack of playing cards consisting of 10
hearts, 8 spades, and 7 clubs. If she selects a card at
random from this stack, what is the probability that it
is a heart or a club?
These are mutually exclusive events since the card
cannot be both a heart and a club. Note there is a total of
25 cards.
Mutually
exclusive events
Substitute
and add.
Answer: The probability that Sylvia selects a heart or a
club is
Sylvia has a stack of playing cards consisting of 10
hearts, 8 spades, and 7 clubs. If she selects a card at
random from this stack, what is the probability that it
is a spade or a club?
Answer:
The Film Club makes a list of 9 comedies and 5
adventure movies they want to see. They plan to
select 4 titles at random to show this semester. What
is the probability that at least two of the films they
select are comedies?
At least 2 comedies means that the selected movies may
include 2, 3 or 4 comedies. It is not possible to select 2
comedies, 3 comedies, and 4 comedies all in the same
semester, so the events are mutually exclusive. Add the
probabilities of each type of movie.
P(2)
2 comedies,
2 adventures
P(3)
3 comedies,
1 adventure
P(4)
4 comedies,
0 adventures
Simplify.
Answer: The probability that at least two of the films are
comedies is
or about 0.91.
The Book Club makes a list of 9 mysteries and 3
romance books they want to read. They plan to select
3 titles at random to read this semester. What is the
probability that at least two of the books they select
are romances?
Answer:
or about 0.04
There are 2400 subscribers to an Internet service
provider. Of these, 1200 own Brand A computers, 500
own Brand B, and 100 own both A and B. What is the
probability that a subscriber selected at random owns
either Brand A or Brand B?
Since some subscribers own both A and B, the events
are inclusive.
P(A)
P(B)
P(A or B) = P(A) + P(B) – P(A and B)
Substitute and simplify.
Answer: The probability that a subscriber owns
either A or B is
There are 200 students taking Calculus, 500 taking
Spanish, and 100 taking both. There are 1000
students in the school. What is the probability that a
student selected at random is taking Calculus or
Spanish?
Answer:
Example 1 Choose a Measure of Central Tendency
Example 2 Standard Deviation
A new Internet company has 3 employees who are
paid $300,000, 10 who are paid $100,000, and 60 who
are paid $50,000.
Which measure of central tendency best represents
the pay at this company?
Since most of the employees are paid $50,000, the higher
values are outliers.
Answer: Thus, the median or mode best represents the
available jobs.
A new Internet company has 3 employees who are
paid $300,000, 10 who are paid $100,000, and 60 who
are paid $50,000.
Which measure of central tendency would recruiters
for this company most likely use to attract
job applicants?
Answer: The recruiters would use the mean ($67,123)
to make it look like the employees would make
more money.
In a cereal contest, there is 1 Grand Prize of
$1,000,000, 10 first prizes of $100, and 50 second
prizes of $10.
a. Which measure of central tendency best represents
the prizes?
Answer: median or mode
b. Which measure of central tendency would advertisers
be most likely to use?
Answer: mean
Rivers This table shows the length in thousands of
miles of some of the longest rivers in the world. Find
the standard deviation for these data.
River
Length
(thousands of miles)
Nile
Amazon
Missouri
Rio Grande
4.16
4.08
2.35
1.90
Danube
1.78
Step 1
Find the mean. Add the data and divide by
the number of items.
The mean is about 2.85 thousand miles.
Step 2
Find the variance.
Variance formula
5
Simplify.
The variance is about 1.104 thousand
miles or 1,104 miles.
Step 3
Find the standard deviation.
Take the square root of each side.
Answer: The standard deviation is about 1.05
thousand miles.
A teacher has the following test scores: 100, 4, 76, 85,
and 92. Find the standard deviation for these data.
Answer: 34.62
Example 1 Classify a Data Distribution
Example 2 Normal Distribution
Determine whether the data {31, 33, 37, 35, 33, 36, 34,
36, 32, 36, 33, 32, 34, 34, 35, 34} appear to be
positively skewed, negatively skewed, or normally
distributed.
Make a frequency table for the data. Then use the
table to make a histogram.
Value
Frequency
31
1
32
2
33
3
34
4
35
2
36
3
Answer: Since the data are somewhat
symmetric, this is a normal distribution.
Determine whether the data {7, 5, 6, 7, 8, 4, 6, 8, 7,
6, 6, 4} appear to be positively skewed, negatively
skewed, or normally distributed.
Answer: negatively skewed
Students counted the number of candies in 100 small
packages. They found that the number of candies per
package was normally distributed with a mean of 23
candies per package and a standard deviation of 1
piece of candy.
About how many packages had between 24 and
22 candies?
Draw a normal
curve. Label the
mean and positive
and negative
multiples of the
standard deviation.
The values of 22 and 24 are 1 standard deviation below
and above the mean, respectively. Therefore, 68% of the
data are located here.
Multiply 100 by 0.68.
Answer: About 68 packages
contained between 22 and
24 pieces.
What is the probability that a package selected at
random had more than 25 candies?
The value 25 is two standard deviations above the mean.
You know that about 100% – 95% or 5% of the data are
more than two standard deviations away from the mean.
By the symmetry of the normal curve, half of 5%, or
2.5%, of the data are more than two standard deviations
above the mean.
Answer: The probability that a package selected at
random had more than 25 candies is about 2.5%
or 0.025.
Students counted the number of candies in 100 small
packages. They found that the number of candies per
package was normally distributed with a mean of 23
candies per package and a standard deviation of 1
piece of candy.
a. About how many packages had between 25 and
21 candies?
Answer: 95
b. What is the probability that a package selected at
random had greater than 24 candies?
Answer: 16% or 0.16
Example 1 Binomial Theorem
Example 2 Binomial Experiment
If a family has 4 children, what is the probability that
they have 2 girls and 2 boys?
There are two possible outcomes for the gender of each
of their children: boy or girl. The probability of a boy
and the probability of a girl
The term
represents 2 girls and 2 boys.
Multiply.
Answer: The probability of 2 boys and 2 girls
is
If a family has 4 children, what is the probability that
they have 4 boys?
Answer: The probability of 4 boys is
.
A report said that approximately 1 out of 6 cars sold
in a certain year was green. Suppose a salesperson
sells 7 cars per week.
What is the probability that this salesperson will sell
exactly 3 green cars in a week?
The probability that a sold car is green is
The probability that a sold car is not green is
There are C(7, 3) ways to choose the three green cars
that sell.
If he sells three
green cars, he
sells four that
are not green.
Simplify.
Answer: The probability that he will sell at least 3 green
cars is
or about 0.096.
A report said that approximately 1 out of 6 cars sold
in a certain year was green. Suppose a salesperson
sells 7 cars per week.
a. What is the probability that this salesperson will sell
exactly 4 green cars in a week?
Answer: 0.016
b. What is the probability that this salesperson will sell
at least 2 green cars in a week?
Answer: 0.33
Example 1 Biased and Unbiased Samples
Example 2 Find a Margin of Error
Example 3 Analyze a Margin of Error
State whether the following method would produce a
random sample. Explain.
surveying people going into an action movie to find
out the most popular kind of movie
Answer: No; they will most likely think that action
movies are the most popular kind of movie.
State whether the following method would produce a
random sample. Explain.
calling every 10th person on the list of subscribers to
the newspaper to ask about the quality of service
Answer: Yes; no obvious bias exists in calling every
10th caller.
State whether each method would produce a random
sample. Explain.
a. surveying people going into a football game to find out
the most popular sport
Answer: No; they will most likely think that football is the
most popular kind of sport.
b. surveying every fifth person going into a mall to find
out the most popular kind of movie
Answer: Yes; no obvious bias exists in asking every
5th person.
In a survey of 100 randomly selected adults, 37%
answered “yes” to a particular question. What is the
margin of error?
Formula for margin of
sampling error
Use a calculator.
Answer: The margin of error is about 10%. This means
that there is a 95% chance that the percent of
people in the whole population who would
answer “yes” is between
and
In a survey of 100 randomly selected adults, 50%
answered “no” to a particular question. What is the
margin of error?
Answer: 10%
Health In an earlier survey, 30% of the people
surveyed said they had smoked cigarettes in the past
week. The margin of error was 2%.
What does the 2% indicate about the results?
Answer: There is a 95% chance that the percent of
people in the population who had smoked
cigarettes in the past week was between
28% and 32%.
Health In an earlier survey, 30% of the people
surveyed said they had smoked cigarettes in the past
week. The margin of error was 2%.
How many people were surveyed?
Formula for margin
of sampling error
Divide each side
by 2.
Square each side.
Multiply by n and divide
by 0.0001.
Use a calculator.
Answer: About 2100 people were surveyed.
Health In an earlier survey, 25% of the people
surveyed said they had exercised in the past week.
The margin of error was 2%.
a. What does the 2% indicate about the results?
Answer: There is a 95% chance that the percent of
people in the population that had exercised in
the past week was between 23% and 27%.
b. How many people were surveyed?
Answer: 1875
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