Download Algebra 2 Notes

Algebra 2 Notes
Objective(s):
Section 10.1: Apply the Counting Principle and Permutations
Use the fundamental counting principle and permutations.
Vocabulary:
I. Fundamental Counting Principle:
Two Events:
If one event can occur in m ways and another event can occur in n ways, then
the number of ways that both events can occur is m  n.
Three or more
Events:
The fundamental counting principle can be extended to three or more
events. If three events can occur in m, n, and p ways, then the number of
ways that all three events can occur is m  n  p.
II. Permutation:
(top of p. 684)
An ordering of n objects.
III. Factorial:
(see Glossary)
For any positive integer n, the expression n!, read "n factorial," is the
product of all the integers from 1 to n. Also, 0! is defined to be 1.
n! = n (n ‒1)  (n ‒ 2)     3  2  1
IV. Permutations of n Objects
Taken r at a Time:
The number of permutations of r objects taken from a group
of n distinct objects is denoted by nPr and is given by the formula:
nPr
V. Permutations with Repetition:
=
n!
(n ‒ r)!
The number of distinguishable permutations of n objects where
one object is repeated s1 times, another is repeated s2 times,
n!
and so on is: s1! s2!    sk!
Notes 10.1 page 2
Examples:
1. At a blood drive, blood can be labeled one of four types (A, B, AB, or O),
one of two Rh factors (+ or ‒), and one of two genders (F or M). How many
different ways can blood be labeled?
16
2. At a used book sale, you are interested in 5 novels, 3 books of nonfiction, and
7 comic books. If you buy one of each kind, how many different choices do you
have?
105
3. The digits 0, 1, 2, 3, and 4 are used to generate four-digit customer codes.
How many different codes are possible if digits
a. can be repeated?
625
b. cannot be repeated?
120
4. A license plate consists of two digits followed by four letters. How many different license plates can be made
if the letters O and I cannot be used ?
33,177,600
5. A television news director has 8 news stories to present on the evening news.
a. How many different ways can the stories be presented?
40,320
b. If only 3 of the stories will be presented, how many possible ways can a
lead story, a second story, and a closing story be presented?
336
6. How many different ways can 4 raffle tickets be selected from 50 tickets if each
ticket wins a different prize?
5,527,200
7. Find the number of distinguishable permutations of the letters in the following:
a. CARROLL
b. RAPP
c. SPAGNUOLO
1260
12
181,440
Algebra 2 Notes
Objective(s):
Section 10.2: Use Combinations and the Binomial Theorem
Use combinations and the binomial theorem.
Vocabulary:
I. Combination:
A selection of r objects from a group of n objects where the order is
not important.
II. Combinations of n Objects
Taken r at a Time:
The number of combinations of r objects taken from a group
of n distinct objects is denoted by nCr and is given by the
formula:
nC r
III. Multiple Events:
=
When finding the number of ways
n!
(n ‒ r)!  r!
both an event A and an event B can
occur, you need to multiply
. When finding the number of ways
that
event A or event B can occur, you must add instead.
IV. Subtracting Possibilities: Counting problems that involve phrases like
are sometimes easier to solve by
"at least"
subtracting
.
or
"at most"
possibilities you
do not
want from the total number of possibilities.
V. Pascal's Triangle:
Pascal's triangle is shown below with its entries represented by
combinations and with its entries represented by numbers. The first
and last numbers in each row are 1. Every number other than 1 is the
sum of the closest two numbers in the row directly above it.
.
Notes 10.2 page 2
VI. Binomial Theorem:
For any positive integer n, the binomial expansion of (a + b)n is:
(a + b)n = nC0anb0 + nC1an ‒ 1b1 + nC2an ‒ 2b2 +  + nCna0bn
Notice that each term in the expansion of (a + b)n has the form
n ‒ r br
nC r a
where r is an integer from 0 to n.
Examples:
1. Parents have 10 books that they can read to their children this week.
Five of the books are nonfiction and 5 are fiction.
a. If the order in which they read the books is not important, how many
different sets of 4 books can they choose?
210
b. In how many groups of 4 books are all the books either nonfiction or fiction?
10
2. The Student Senate consists of 6 seniors, 5 juniors, 4 sophomores, and 3 freshmen.
a. How many different committees of exactly 2 seniors and 2 juniors can be chosen?
150
b. How many different committees of at most 4 students can be chosen?
4,047
3. You are going to toss 10 different coins. How many different ways will at least 4 of the coins show heads?
848
Notes 10.2 page 3
4. From a collection of 7 baseball caps, you want to trade 3. Use Pascal's triangle
to find the number of combinations of 3 caps that can be traded.
35
5. Use the binomial theorem to write the binomial expansion of (x + y) 6.
x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + y6
6. Use the binomial theorem to write the binomial expansion of (3x ‒ 2)4.
81x4 ‒ 216x3 + 216x2 ‒ 96x + 16
7. Find the coefficient of x3y4 in (2x ‒ y)7.
280
Algebra 2 Notes
Objective(s):
Section 10.3: Define and Use Probability
To find the likelihood that an event will occur.
Vocabulary:
I. Probability:
A number from 0 to 1 that indicates the likelihood the event will
occur. Probabilities can be written as fractions, decimals, or percents.
or percents.
II. Theoretical Probability
of an Event:
When all outcomes are equally likely, the theoretical probability that
an event A will occur is: P(A) =
Number of outcomes in event A
Total number of outcomes
The theoretical probability of an event is often simply called the
probability of the event.
III. Odds:
A measure of the chances in favor of an event occurring or
the chances against an event occurring.
IV. Odds in Favor of or
Odds Against an Event:
When all outcomes are equally likely, the odds in favor of an event
A and the odds against an event A are defined as follows:
Odds in favor of event A =
Odds against event A =
Number of outcomes in A
Number of outcomes not in A
Number of outcomes not in A
Number of outcomes in A
a
You can write the odds in favor of or against an event in the form b
or in the form a:b.
V. Experimental Probability
of an Event:
When an experiment is performed that consists of a certain number
of trials, the experimental probability of an event A is given by:
P(A) =
Number of trials where A occurs
Total number of trials
Notes 10.3 page 2
VI. Geometric probabilities:
Probabilities found by calculating a ratio of two lengths, areas, or
volumes.
Examples:
1. You pick a card from a standard deck of 52 playing cards. Find the probability of
a. picking an 8
4
=
52
1
13
b. picking a red king
2
=
52
1
26
2. A cereal company plans to put 5 new cereals on the market: a wheat cereal, a rice cereal,
a corn cereal, an oat cereal, and a multigrain cereal. The order in which the cereals are
introduced will be randomly selected. Each cereal will have a different price.
a. What is the probability that the cereals are introduced in order of their
suggested retail price?
1
120
≈ 0.0083
b. What is the probability that the first cereal introduced will be the multigrain cereal?
1
5
or 0.2
Notes 10.3 page 3
3. A standard six-sided die is rolled. Find
a. the odds in favor of rolling a 6.
1:5
b. the odds against rolling an odd number.
1:1
4. The blood types for a sample of donors at a blood drive are displayed
in the bar graph. Find the experimental probability that a randomly
selected blood donor would have blood type O.
0.44
5. Find the probability that a dart thrown at the rectangular board
hits one of the triangles. Assume that the dart is equally likely to
hit any point inside the board.
0.08
Algebra 2 Notes
Objective(s):
Section 10.4: Find Probabilities of Disjoint and Overlapping Events
You will find probabilities of compound events.
Vocabulary:
I. Compound event:
The union or intersection of two events.
II. Overlapping events:
Two events that have one or more outcomes in common.
III. Disjoint/mutually
exclusive:
Two events that have no outcomes in common.
IV. Probability of Compound
Events:
If A and B are any two events, then the probability of A or B is:
P(A or B) = P(A) + P(B) − P(A and B)
If A and B are disjoint events, then the probability of A or B is:
P(A or B) = P(A) + P(B)
V. Complement events:
The event A that consists of all outcomes that are not in A.
The notation A is read as "A bar."
VI. Probability of the
Complement of an
Event:
The probability of the complement of A is P(A) = 1 − P(A)
Examples:
1. A six-sided die is rolled. What is the probability that the number rolled is less than 3 or greater than 5?
1
2
2. A six-sided die is rolled. What is the probability of rolling a number greater than 4 or even?
2
3
Notes 10.4 page 2
3. Of 100 students surveyed, 92 own either a car or a computer. Also, 65 own cars and 82 own computers.
What is the probability that a randomly selected student owns both a car and a computer?
0.55
4. A card is drawn from a standard deck of 52 cards. Find the probability of each event.
a. The card is not a 7.
12
or ≈ 0.923
13
b. The card is not a red face card.
46
52
or ≈ 0.885
5. Two six-sided dice are rolled. What is the probability that the sum is neither 4 nor 8?
7
9
or ≈ 0.778
Algebra 2 Notes
Objective(s):
Section 10.5: Find Probabilities of Independent and Dependent Events
To examine independent and dependent events.
Vocabulary:
I. Independent
Events:
Two events in which the occurrence of one has no effect on the occurrence of
II. Probability of
Independent
Events:
If A and B are independent events, then the probability that both A and B
the other.
occur is:
P(A and B) = P(A) • P(B)
More generally, the probability that n independent events occur is the
product of the individual events.
III. Dependent
Events:
Two events A and B where the occurrence of one affects the occurrence of the
IV. Conditional
Probability:
The probability that B will occur given that A has occurred and is written as
V. Probability of
Dependent
Events:
If A and B are dependent events, then the probability that both A and B occur
other.
P(BǀA).
is:
P(A and B) = P(A) • P(BǀA).
• (top of p. 720) The formula for finding probabilities of dependent events
can be
extended to three or more events.
Examples:
1. In a survey at a football game, 50 of 75 male fans and 40 of 50 female fans said that they favor the new team mascot.
If 1 male and 1 female fan are randomly selected, what is the probability that both favor the new mascot?
8
15
Notes 10.5 page 2
2. A survey found that 46% of parents surveyed say that they
read to their children at least once a week. If 3 parents are
selected at random, what is the probability that all 3 will say
that they read to their children at least once a week?
≈ 0.097
3. During each of the 5 days of a particular week, an employee is
randomly given 1 of 10 prizes. All prizes are available each day,
and one of the prizes is a $500 gift certificate. What is the probability
that an employee receives the $500 prize at least once?
≈ 0.41
4. The table shows the status of 200 registered college students.
Part Time
Full Time
Female
80
40
Male
60
20
a. What is the probability that a randomly selected student is female?
3
5
= 0.6
b. What is the probability that a randomly selected student if female, is a full time student?
2
5
= 0.4
Notes 10.5 page 3
5. You randomly select 2 cards from a standard deck of 52 cards.
What is the probability that the first card is a heart and the
second is a club if
a. you replace the first card before selecting the second.
1
16
= 0.0625
b. you do not replace the first card before selecting the second.
13
204
≈ 0.064
6. Suppose your area has 8 different Internet Service Providers (ISPs) and you and 3 friends randomly select your own ISP.
What is the probability that you all choose different ISPs?
≈ 0.41
7. On a manufacturing line, 20% of all the items produced are defective. Although all the items are inspected before they are
shipped, 10% of the items are incorrectly classified as either defective or not defective. What percent of the items will be
classified as not defective?
74%
Algebra 2 Notes
Objective(s):
Section 11.1: Find Measures of Central Tendency and Dispersion
To describe data using statistical measures.
Vocabulary:
I. Statistics:
Numerical values used to summarize and compare sets of data.
II. Measure of Central
Tendency (definition):
A number used to represent the center or middle of a set of data
III. Measures of Central
Tendency:
•
values.
The mean, or average, of n numbers is the sum of the numbers
divided by n. The mean is denoted x which is read as "x-bar." For the
x1 + x 2 +    + x n
x=
n
data set x1, x2, . . . , xn, the mean is
•
The median of n numbers is the middle number when the numbers
are written in order. (If n is even, the median is the mean of the two
middle numbers.)
•
The mode of n numbers is the number or numbers that occur most
frequently. There may be one mode, no mode, or more than
one mode.
IV. Measure of dispersion:
A statistic that tells you how dispersed, or spread out, data values are.
Examples: Range, standard deviation.
V. Range:
The difference between the greatest and least data values.
VI. Standard deviation:
(definition)
The typical difference (or deviation) between a data value and the
mean.
Notes 11.1 page 2
VII. Standard Deviation
of a Data Set:
(Key Concept)
The standard deviation σ (read as "sigma") of x1, x2, . . . , xn is :
(x1 − x)2 + (x2 − x)2 +    + (xn − x)2
n
VIII. Outlier:
A value that is much greater than or much less than most of the other
values in a data set.
Examples:
1. The data sets give the times in minutes for runners in two races.
Find the mean, median, and mode(s) of each data set.
Race A
Race B
4, 7, 8, 8, 9, 9,
10, 11, 12
5, 6, 6, 7, 8, 8, 8,
10, 11
Race A:
mean ≈ 8.7
median = 9
modes = 8 and 9
Race B:
mean ≈ 7.7
median = 8
mode = 8
2. Find the range of the times in each data set of Example #1.
Race A: 8
Race B: 6
3. What is the standard deviation of the times in each data set in Example #1?
2.2 and 1.8
Notes 11.1 page 3
4. You are training for a triathlon. The miles that you ride your bicycle for 7 weeks are: 17, 20, 16, 18, 22, 19, 20.
a. Find the mean, median, mode, range, and standard deviation of the data set.
mean: 18.9
median: 19
mode: 20
range: 6
st. dev: 1.9
b. Your mileage the next week is an outlier, 3. Find the new mean, median, mode, range, and standard deviation.
mean: 16.9
median: 18.5
mode: 20
range: 19
st. dev: 5.2
c. Which measure of central tendency does the outlier effect the most? the least?
mean; mode
d. What effect does the outlier have on the range and standard deviation?
The outlier causes both the range and standard deviation to increase.
Algebra 2 Notes
Objective(s):
Section 11.2: Apply Transformations to Data
To learn how transformations of data affect statistics.
Vocabulary:
I. Adding a Constant
to Data Values:
When a constant is added to every value in a data set, the following are
true:
•
The mean, median, and mode of the new data set can be
obtained by adding the same constant to the mean, median,
and mode of the original data set.
•
II. Multiplying Data Values
by a Constant:
The range and standard deviation are unchanged.
When each value of a data set is multiplied by a constant, the new
mean, median, mode, range, and standard deviation can be found by
multiplying each original statistic by the same constant.
Notes 11.2 page 2
Examples:
1. The following data give the weights of 5 people: 138, 142, 155, 140, 155.
At the end of a month, each person had lost 3 pounds. Give the mean, median,
mode, range, and standard deviation of the starting weights and the weights
at the end of the month.
Mean:
Median:
Mode:
Range:
Std. dev:
Starting
146
142
155
17
7.46
End of month
143
139
152
17
7.46
2. Give the mean, median, mode, range, and standard deviation for the starting weights
of the 5 people in Example 1 in kilograms (Note: 1 kilogram = 0.45 pound)
Mean: 65.7
Median: 63.9
Mode: 69.75
Range: 7.65
Std. dev: 3.36
3. The data set below gives the number of field goals made by the 10 kickers who kicked the most field goals
during the 2004 National Football League regular season.
31, 29, 29, 28, 27, 27, 25, 24, 24, 24
a. Find the mean, median, mode, range, and standard deviation of the number of field goals made.
Mean: 26.8
Median: 27
Mode: 24
Range: 7
Std. dev: ≈ 2.4
b. A field goal is worth three points. Find the mean, median, mode, range, and standard deviation of the number of
points scored on field goals by the 10 kickers.
Mean: 80.4
Median: 81
Mode: 72
Range: 21
Std. dev: ≈ 7.1