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palindromic numbers

#'s that are palindromes

Using the 36 possibilities found in the product table for rolling two dice, list and count the outcomes for which the sum (for both dice) is the following. Multiple of 3

(1,2), (2,1), (2,4), (4,2), (1,5), (5,1), (3,3), (3,6), (6,3), (4,5), (5,4), (6,6); 12

Find the measure of each marked angle. Assume the lines are parallel.

(2x - 24)` = 34` (x+5)` = 34`

Using the 36 possibilities found in the product table for rolling 2 dice, list and count the outcomes for which the sum (for both dice) is the following. Greater than 10

(6,5),(5,6),(6,6): 3

Proof of Formulas

(a) P(A and B)=P(A)⋅P(B|A) (b) Therefore, P(B|A)=P(A and B)P(A). (c) Therefore, P(B|A)=n(A and B)/n(S)n(A)/n(S). (d) Therefore, P(B|A)=n(A and B)n(A).

permutation

(arrangements) Permutations involve the number of arrangements of n things taken r at a time, where repetitions are not allowed.; >> nPr >> must satisfy these conditions: 1. repetitions are not allowed; 2. order is important

Use subset (equals or is not a subset of) in the blank to make a true statement. {10, 29, 34} ______ {5, 29, 34, 44} (is not a subset of)

(equals or is not a subset of)

law of large numbers

(law of averages)A theoretical probability really says nothing about one, or even a few, repetitions of an experiment, but only about the proportion of successes we would expect over the long run. >>> As an experiment is repeated more and more times, the proportion of outcomes favorable to any particular eveny will tend to come closer and closer to the theoretical probability of that event

connectives - 84

(logical connectives) such as *and, or, not, if,... then, can be used in forming compound statements

expected value

(mathematical expectation) the quantity expected If a random variable x can have any of the values x1,x2,x3,...,xn, and the corresponding probabilities of these values occurring are P(x1),P(x2),P(x3),...,P(xn), then E(x), the expected value of x , is calculated as follows. E(x)=x1⋅P(x1)+x2⋅P(x2)+x3⋅P(x3) + ⋯+xn⋅P(xn)

combination

(subset) the number of size-r subsets, given a set of size n—is written nCr; There are n things available and we are choosing r of them, so we can read nCr as "n choose r"

truth value - 84

(the truth or falsity) of statements with multiple parts. The truth value of such statements depends on their components.

universal qualifiers - 86

*indicate all members* all, each, every, no(ne) be careful when forming the negation of a statement involving quantifiers

Suppose you buy 1 ticket for $1 out of a lottery of 1,000 tickets where the prize for the one winning ticket is to be $500. What is your expected net winnings?

-0.50

Suppose a charitable organization decides to raise money by raffling a trip worth $500. If 3,000 tickets are sold at $1.00 each, find the expected net winnings for a person who buys 1 ticket. Round to the nearest cent.

-0.83

Mendel found that flower color in certain pea plants obeyed the following scheme. Pure red crossed with pure white produces red. When pure red (RR) and pure white (rr) parents are crossed, the resulting Rr combination (one of each gene) produces second generation offspring with red flowers, since red is dominant. Suppose one of these second generation Rr flowers is crossed with a pure red. What is the probability that the resulting plant will have white flowers?

Zero factorial

0! = 1

Factorials Table 0 - 10

0! = 1 1! = 1 2! = 2 3! = 6 4! = 24 5! = 120 6! = 720 7! = 5040 8! = 40,320 9! = 362,880 10! = 3,628,800

For n repeated independent trials, with constant probability of success p for all trials, find the probability of exactly x successes. Round your answer to four decimal places. n=15, p=1/6, x=7

0.0053

A student takes a true-false test consisting of 14 questions. Assuming that the student guesses at each question, find the probability that the student answers exactly 12 questions correctly. Round your answer to four decimal places as needed.

0.0056

A student takes a true-false test consisting of 11 questions. Assuming that the student guesses at each question, find the probability that the student answers exactly 9 questions correctly. Round your answer to four decimal places as needed.

0.0269

For n repeated independent trials, with constant probability of success p for all trials, find the probability of exactly x successes. Round your answer to four decimal places. n=13, p=1/8, x=4

0.0525

In one town, 25% of all voters are Democrats. If two voters are randomly selected for a survey, find the probability that they are both Democrats. Round your answer to three decimal places as needed.

0.063

In one town, 31% of all voters are Democrats. If two voters are randomly selected for a survey, find the probability that they are both Democrats. Round your answer to three decimal places as needed.

0.096

What is the probability that 18 tosses of a fair coin will show 11 tails? Round the answer to four decimal places as needed.

0.1214

Find the probability that when a gardener plants 20 seeds, she harvests 16 radishes given the probability that a radish seed will germinate is 0.7. Round the answer to three decimal places as needed.

0.130

What is the probability that 19 tosses of a fair coin will show 9 tails? Round the answer to four decimal places as needed.

0.1762

What is the probability that 12 tosses of a fair coin will show 7 tails? Round the answer to four decimal places as needed.

0.1934

In one town, 48% of all voters are Democrats. If two voters are randomly selected for a survey, find the probability that they are both Democrats. Round your answer to three decimal places as needed.

0.23

A student from the community college is selected at random. Find the probability that the student is between 26 and 35 inclusive. Round approximations to three decimal places. Age (years)Number of students Under 21 409 21-25 409 26 -30 208 31-35 57 Over 35 25 Total 1,108

0.239

A student from the community college is selected at random. Find the probability that the student is between 26 and 35 inclusive. Round approximations to three decimal places. Age (years)Number of students Under 21 401 21-25 413 26 -30 213 31-35 54 Over 35 23 Total 1,104

0.242

For n repeated independent trials, with constant probability of success p for all trials, find the probability of exactly x successes. Round your answer to four decimal places. n=16, p=1/6, x=3

0.2423

A student from the community college is selected at random. Find the probability that the student is between 26 and 35 inclusive. Round approximations to three decimal places. Age (years)Number of students Under 21 407 21-25 408 26 -30 220 31-35 53 Over 35 20 Total 1,108

0.246

A batch of 100 calculators contains 5 defective calculators. If 6 calculators are selected at random from this batch, determine the probability that exactly two of those selected are defective. Round your answer to four decimal places as needed.

0.267

A bag contains 12 red chips and 9 blue chips. Two chips are selected randomly without replacement from the bag. What is the probability that the two chips are the same color? Round your answer to three decimal places as needed.

0.486

A bag contains 8 red chips and 5 blue chips. Two chips are selected randomly without replacement from the bag. What is the probability that the two chips are the same color? Round your answer to three decimal places as needed.

0.487

The table below describes the smoking habits of a group of asthma sufferers. Nonsmoker - Light smoker - Heavy smoker - Total Men - 324 - 76 - 77 - 477 Woman - 346 - 75 - 76 - 497 Total - 670 - 151 - 153 - 974 If one of the 974 subjects is randomly selected, find the probability that the person chosen is a woman given that the person is a light smoker. Round your answer to three decimal places as needed.

0.497

The table below describes the smoking habits of a group of asthma sufferers. Nonsmoker - Light smoker - Heavy smoker - Total Men - 348 - 83 - 66 - 497 Woman - 376 - 82 - 65 - 523 Total - 724 - 165 - 131 - 1,020 If one of the 974 subjects is randomly selected, find the probability that the person chosen is a woman given that the person is a light smoker. Round your answer to three decimal places as needed.

0.497

A bag contains 8 red chips and 4 blue chips. Two chips are selected randomly without replacement from the bag. What is the probability that the two chips are the same color? Round your answer to three decimal places as needed.

0.515

The table below describes the smoking habits of a group of asthma sufferers. Nonsmoker - Light smoker - Heavy smoker - Total Men - 318 - 76 - 80 - 474 Woman - 386 - 85 - 74 - 512 Total - 671 - 161 - 154 - 986 If one of the 986 subjects is randomly selected, find the probability that the person chosen is a woman given that the person is a light smoker. Round your answer to three decimal places as needed.

0.528

The distribution of B.A. degrees conferred by a local college is listed below, by major. What is the probability that a randomly selected degree is not in Mathematics? Round your answer to three decimal places as needed. Major Frequency English 2073 Mathematics 2164 Chemistry 318 Physics 856 Liberal Arts 1358 Business 1676 Engineering 868 Total 9313

0.768

A batch consists of 12 defective coils and 88 good ones. Find the probability of getting two good coils when two coils are randomly selected if the first selection is replaced before the second is made.

0.7744

The table shows the distribution of family size in a certain U.S. city. A family is selected at random from the city. Find the probability that the size of the family is between 2 and 5 inclusive. Round approximations to three decimal places. Family Size Probability 2 0.397 3 0.23 4 0.203 5 0.117 6 0.037 7+ 0.016

0.947

0.948

What is the probability that 4 randomly selected people all have different birthdays? Round your answer to four decimal places.

0.9836

outlier 12.2

1) A piece of data that is quite separated from the rest of the data 2) should usually be considered as possible errors in the data 3) A value much greater or much less than the others in a data set

weighting factor 12.2

1) the number of units in 2) a mathematical factor used to make a disproportionate sample representative.

Two distinct even numbers are selected at random from the first ten even numbers greater than zero. What is the probability that the sum is 30?

1/15

In an essay contest, a teacher finds that seven students have written excellent essays. Three of these students are Alicia, Pat, and David. If the teacher chooses the first place winner, second place winner, and third place winner at random from these seven students, what is the probability that Alicia will win first prize, Pat will win second prize, and David will win third prize?

1/210

Three fair coins are tossed. Find the probability of getting the same thing on all three coins.

1/4

When two balanced dice are rolled, there are 36 possible outcomes. What is the probability that the sum of the numbers on the dice is 6 or 9?

1/4

Give the probability that the spinner shown would land on the color white.

1/6

If two fair dice are rolled, find the probability that the sum is 6 given that the roll is a "double".

1/6

Four boys and three girls are seated in a row, at random, to watch a play. What is the probability that a girl is seated at each end of the row?

1/7

After rolling the first ball of a frame in a game of 10-pin bowling, how many different pin configurations can remain (assuming all configurations are physically possible)?

1024 (8x8x8x2)

Determine the number of figures (of any size) in the design. Squares (of any size)

The shape of a barbecue pit is like a parallelogram with height of 58 in. and a base of 50 in. It costs $ 0.60 per ft squared to fill the pit with charcoals. What is the total cost?

12.08

class width

12.1 1) for the distribution is the difference of any two successive lower class limits 2) or of any two successive upper class limits

descriptive statistics

12.1 collecting, organizing, summarizing, & presenting data (information)

inferential statistics

12.1 drawing inferences or conclusions (making conjectures) about populations on the basis of information from samples

raw data

12.1 information that has been collected but not yet organized of processed

sample

12.1 some but ordinarily not all, of the items in population

expected frequency distribution

12.1 the 1st 2 columns of table

upper class limits

12.1 the largest unit in a class of class (10-19) upper class limit is 19

observed (empirical) frequencies

12.1 the results from the 1st 2 columns

lower class limits

12.1 the smallest possible data values within the respective classes of class (10-19) lower class limit is 10

expected (theoretical) frequencies

12.1 uses binomial probability formula

grouped frequency distribution

12.1 1. Make sure each data item will fit into one, and only one, class. 2. Try to make all classes the same width. 3. Make sure the classes do not overlap. 4. Use from 5 to 12 classes. Too few or too many classes can obscure the tendencies in the data.

bar graph

12.1 A frequency distribution of non-numerical observations

population

12.1 all items of interest; * includes sample *

line graph

12.1 demonstrates how a quantity changes >> use a line to connect points

circle graph (pie chart)

12.1 graphic alternative to the bar graph

histogram

12.1 a series of rectangles whose lengths represent the frequencies and placed next to one another

stem-and-leaf display

12.1 a tool of exploratory data analysis

classes

12.1 data sets

class mark

12.1 middle value

frequency distribution

12.1 organized data set that includes many repeated items 1) distinct data value (x) 2) with their frequencies (f)

qualitative (non-numerical) data

12.1 page 645

quantitative (numerical) data

12.1 page 645

ranked data

12.1 quantitative data arranged in numerical order

frequency polygon classes

12.1 simple plot a single point at the appropriate height for each frequency, connect the point with a series of connected & complete the polygon with segments that trail down to the axis

relative frequency distribution

12.1 the fraction, or % of data set represented by the item 1) If n denotes the total number of items, and a given item, x, occurred f times, then the relative frequency of x is f/n.

measure of central tendency

12.2 the middle value of the set

How many integers between 100 & 400 contain the digit 2?

138

Determine the number of triangles (of any size) in the figure.

The following figure has perimeter, circumference, or area as indicated. Find the value of x. Use 3.14 as an approximation for pi P= 58

14.5

Refer to the given figure, an isosceles triangle with AB equals AC . Triangle A B C, where side B C is horizontal and vertex A is above side B C. If angle C measures 16.5 degrees find the measure of angle A.

147

How many of the numbers from 10 through 88 have the sum of their digits equal to a perfect square?

15 90=16

Refer to the given figure, an isosceles triangle with AB=AC angle C measures 15 degrees find angle A

150 degrees

Triangle CAB is similar to triangle CSR: Given that SC=60; AS=75; CR=68; find CB???

153

triangle CAB is similar ro triangle csr given sc=60; as=75; cr=68; find cb

153

One of the values r (radius), d (diameter), C (circumference), or A (area) is given for a particular circle. Find the indicated value. Leave pi in your answer. requals 8 cm; Cequals?

16 (pi) cm

Evaluate the factorial expression. n!/r!(n-r)!, n = 38 and r = 9

163,011,640

A sports shop sold tennis rackets in 2 different weights, 3 types of string, and 3 grip sizes. How many different rackets could be sold? A. 12 rackets B. 18 rackets Your answer is correct. C. 8 rackets D. 15 rackets

18 rackets

set equality

2 conditions: Every element of A is an element of B; and; Every element of B is an element of A

A fair die is rolled. What is the probability of rolling an odd number or a number less than 3?

2/3

If 5 apples in a barrel of 25 apples are rotten, what is the expected number of rotten apples in a random sample of 2 apples?

2/5

Mr. Larsen's third grade class has 22 students, 12 girls and 10 boys. Two students must be selected at random to be in the fall play. What is the probability that no boys will be chosen? Order is not important.

2/7

When two balanced dice are rolled, there are 36 possible outcomes. What is the probability that the sum of the numbers on the dice is 6 or 10 ?

2/9

To find the height of this tree, Sarah marked the tree at eye level, 1.8 meters above the ground. She measured 25 m from the base of the tree and then held a 5 cm ruler vertically in front of her eye until the ruler just obscured the tree above the mark. Using a string tied through a hole in one end of the ruler, Sarah found that the distance from her eye to the ruler was 6 cm. What was the height of the tree? Round to the nearest unit.

20 m

Evaluate the factorial expression. 7! / 5!2! exclamation mark EndFraction

Evaluate the expression. 7P3

210

If a single card is drawn from a standard 52-card deck, in how many ways could it be a diamond or a face card?

22 ways

If 2 fair dice, 1 red and 1 white, are rolled, in how many ways can the result be obtained? The sum of the two dice is at least 6.

26 ways

Evaluate 15!/12!

2730

What are the odds against drawing a number greater than 2 from these cards? A group of 5 rectangles in a row are labeled from left to right as follows: 1, 2, 3, 4, 5.

2:3

A basket contains 6 oranges and 4 tangerines. A sample of 3 is drawn. Find the probability that 2 are tangerines and one is an orange.

3/10

A family has three children. What is the probability that two of the children are boys?

3/8

If 2 fair dice, 1 red and 1 white, are rolled, in how many ways can the result be obtained? A different number on each die.

30 ways

Suppose there are 5 roads connecting town A to town B and 6 roads connecting town B to town C. In how many ways can a person travel from A to C via B?

30 ways

A local television station sent out questionnaires to determine if viewers would rather see a documentary, an interview show, or reruns of a game show. There were 650responses with the following results: 195 were interested in an interview show and a documentary comma but not reruns. 26 were interested in an interview show and reruns but not a documentary. 91 were interested in reruns but not an interview show. 156were interested in an interview show but not a documentary. 65 were interested in a documentary and reruns. 39 were interested in an interview show and reruns. 52 were interested in none of the three. How many are interested in exactly one kind of show?

312

A panel containing 5 on-off switches in a row is to be set. Assuming no restrictions on individual switches, use the fundamental counting principle to find the total number of possible panel settings.

One of the values r (radius), d (diameter), C (circumference), or A (area) is given for a particular circle. Find the indicated value. Leave pi in your answer. r=16 cm; C=?

32 pi

A local department store sells carpets in 4 sizes. Each carpet comes in 2 different qualities. One of the sizes comes in 8 colors. The other sizes come in 3 colors. How many choices of carpet are there?

To find the height of this tree, Sarah marked the tree at eye level, 1.8 meters above the ground. She measured 28 m from the base of the tree and then held a 5-cm ruler vertically in front of her eye until the ruler just obscured the tree above the mark. Using a string tied through a hole in one end of the ruler, Sarah found that the distance from her eye to the ruler was 4.4 centimeters. What was the height of the tree? Round to the nearest unit.

Find the area of the circle. Use 3.14 for pi. Round results to two decimal places in necessary. diameter is 21

346.19

4 married couples have reserved 8 seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if no couple is to be separated?

384

Find the area of the trapezoid. top 8 bottom 40 height 16

384

What are the odds in favor of spinning an A on this spinner?

3:5

Counting numbers are to be formed using only the digits 2, 6, and 7 Determine the number of different possibilities for the type of number described below. Four-digit numbers with one pair of adjacent 2s and no other repeated digits (Hint: You may want to split the task of designing such a number into three parts, such as (1) position the pair of 2s, (2) position the 6, and (3) position the 7.)

3x2x1=6

A line segment joins the points (8,14) and (62,227) in the Cartesian plane. Including its endpoints, how many lattice points does this line segment contain? (A lattice point is a point with integer coordinates.)

A basketball player hits three-point shots 42% of the time. If she takes 4 shots during a game, what is the probability that she misses the first shot and hits the last three shots? Round your answer to one decimal place as needed.

4.3%

A basketball player hits three-point shots 44% of the time. If she takes 4 shots during a game, what is the probability that she misses the first shot and hits the last three shots? Round your answer to one decimal place as needed.

4.8%

Evaluate. 17! / 14!

4080

Find the area. Triangle base: 39 height 22 side 25

429 in squared

The following table contains data from a study of two airlines which fly to Small Town, USA. Flights on time Late arrivals Podunk Airlines 33 6 Upstate Airlines 43 5 If one of the 87 flights is randomly selected, find the probability that the flight selected arrived on time given that it was an Upstate Airlines flight.

43/48

43/87

correct, 2.4-2 A local television station sent out questionnaires to determine if viewers would rather see a documentary, an interview show, or reruns of a game show. There were 900 responses with the following results: 270 were interested in an interview show and a documentary comma but not reruns. 36 nbsp were interested in an interview show and reruns but not a documentary. 126 were interested in reruns but not an interview show. 216 were interested in an interview show but not a documentary. 90 were interested in a documentary and reruns. 54 were interested in an interview show and reruns. 72 were interested in none of the three. How many are interested in exactly one kind of show?

432

Evaluate the expression without using a calculator. 32! / 30! * 2!

4961

A line segment joins the points (7,14) and (71,282) in the Cartesian plane. Including its endpoints, how many lattice points does this line segment contain? (A lattice point is a point with integer coordinates.)

Pamela DeMar's computer printer allows for optional settings with a panel of 3 on-off switches in a row. How many different settings can she select if no 2 adjacent switches can both be on?

5.0%

The following table contains data from a study of two airlines which fly to Small Town, USA. Flights on time Late arrivals Podunk Airlines 33 6 Upstate Airlines 43 5 If one of the 87 flights is randomly selected, find the probability that the flight selected, find the probability that the flight selected is an Uptown Airlines flight given that it was late

5/11

Find the area of the parallelogram. length: 10 Height: 5

50 in squared

Suppose triangles ABC and DEF below are similar triangles such that angle Upper B equals 55 degrees and EF overbar=10 cm. Give the measure of angle Upper E

supplement of 124

A line segment joins the points (8,12) and (73,357) in the Cartesian plane. Including its endpoints, how many lattice points does this line segment contain? (A lattice point is a point with integer coordinates.)

Find n(E), given that n(C X E) = 18 and C = {4, 5, 6}.

Evaluate the factorial expression. 13!

6,227,020,800

How many different 8-digit numbers can be written using digits from the set {4, 5, 6, 7, 8} without any repeating digits?

How many two-digit counting numbers are either multiples of 2 or multiples of 3?

60 numbers

Uniform-length matchsticks are used to build a rectangular grid as shown here. If the grid is 11 matchsticks high and 29 matchsticks wide, how many matchsticks are used?

678

The numbers in the Venn Diagram below represent cardinalities. A 15 B 17 C 14 AC 7 BC 4 ABC 2 AB 6 U 21 A Venn diagram with universal set U contains three intersecting circles labeled "A," "B," and "C." Each region is labeled as follows: A only, 15; B only, 17; C only, 14; A and B, 6; A and C, 7; B and C, 4; A, B, and C, 2; U only, 21. Find n(A intersect B' intersect C)

find area parallelogram 12 x 6

A baseball manager has 12 players of the same ability. How many 9 player starting lineups can he create? Round the answer to the nearest whole number.

79,833,600

Find the number of subsets of the set.{13, 14,15}`

Pamela DeMar's computer printer allows for optional settings with a panel of 4 on-off switches in a row. How many different settings can she select if no two adjacent switches can both be on?

refer to the given figure, an isosceles triangle with AB=AC if the perimeter of triangle ABC is 48 and AB=20. what is the lenght of BC?

DeMorgan's Laws to negation of statement 8-1=6 and 20-10 not= 7

8-2 not= 6 or 20-10=7

The shape of a barbecue pit is like a parallelogram with height of 42 in. and a base of 50 in. It costs $ 0.60 per ftsquared to fill the pit with charcoals. What is the total cost?

8.75

Given a group of students: G = {Allen, Brenda, Chad, Dorothy, Eric, Frances, Gale}, count the number of different ways of choosing 4 people for a committee. Assume no one can hold more than one office and that each person is to hold a different position on the committee.

840

Find the requested angle. Supplement of 95 degrees

85 degrees

find the area of triangle height 36 m base 48 m long side 40 m

864 m squared

Evaluate the permutation. 14 P 14

87,178,291,200

find rectangle area a=80 b=110

8800

A bag contains 8 red marbles, 2 blue marbles, and 1 green marble. If a marble is selected at random, what is the probability that it is not blue?

9/11

Evaluate the expression without using a calculator 43! / 41! * 2!

903

Find the area of the circle circumference=11 yd.

94.99 yd squared

Evaluate the expression without using a calculator 45! / 43! * 2!

990

modus ponens - 124

> >>law of detachment

decile 12.4

>> are the nine values (denoted D1, D2,..., D9) along the scale that divide a data set into ten (approximately) equal-sized parts >> We can evaluate deciles by finding their equivalent percentiles.

circle

>>a simple closed curve defined as follows. >>a set of points in a plane, each of which is the same distance from a fixed point

equivalent - 112-113

>conditional statement and its contrapositive are equivalent >converse and inverse are equivalent

For the given sets, construct a Venn diagram and place the elements in the proper region. Let U = {1,2,3,4,5,6,7,8}, A = {3,6,8}, B = {4,6}, C = {1,6,7,8}

A = 3; B = 4; C = 1, 7; AB = 0; ABC = 6; AC = 8; BC = 0; U = 2,5

right circular cone

A cone with circular base having its apex (highest point) directly above the center of its base

plane

A flat surface; surface which lies evenly with the straight lines itself; may be named by 3 Capital Letters representing points that lie in the plane; or by a letter of the Greek alphabet, such as α alpha (alpha), β beta (beta), or γ gamma (gamma).

bi-modal 12.2

A histogram with two peaks (modes)

proper subset of a set

A is a proper subset of set if A B and A B; written as A B

probability distribution

A listing, which shows all possible values of a random variable, along with the probabilities that those values will occur, is a probability distribution for that random variable. All possible values are listed, so they make up the entire sample space, and thus the listed probabilities must add up to 1

discrete random variable

A random variable that can assume only certain fixed values.

empirical probability

A series of repeated experiments provides an empirical probability for an event, which, by "inductive reasoning", is an estimate of the event's theoretical probability. Increasing the number of repetitions increases the reliability of the estimate.

null set

A set with no elements

contraction/ shrink

A size transformation having magnitude k<

dilation/ stretch

A size transformation having magnitude k>1

identity translation

A translation of magnitude 0 leaves every point of the plane unchanged

inverses

A translation of magnitude k, followed by a similar translation of magnitude k but of opposite direction, returns a point to its original position, so these two translations are inverses of each other

Write a description of the shaded region using the symbols A, B, C, union , intersect ,minus and prime as needed.

A' intersect B

Ann's collection of eight albums includes one classical album. Ann will choose four of her albums to play on a road trip. (Assume order is not important.) a) How many different sets of four albums could she choose? b) How many of these sets would not include the classical album? c) How many of them would include the classical album?

A. 70 B. 35 C. 35

Find the unknown side lengths in similar triangles PQR and ABC AC=24 AB=18 BC=30 QR=35 RP=b QP=a

A=21 B=28

A club N with 4 members is shown below: {Alfred, Blake, Carrie, Douglas} Assuming all members of the club are eligible, but no one can hold more than one office, list and count the different ways the club could elect both a president and a treasurer

AB,AC,AD,BA,BC,BD,CA,CB,CD,DA,DB,DC 12

Assuming all members of the club are eligible, but that no one can hold more than one office, list and count the different ways the club could elect a president and a treasurer if the two officers must be the same gender. N = {Alvin, Ben, Carrie, Dennis, Eileen}

AB,AD,BA,BD,CE,DA,DB,EC 8

Assuming all members of the club are eligible, but that no one can hold more than one office, list and count the different ways the club could elect a president and a treasurer if the two officers must not be the same gender. N = {Aaron, Bob, Carla, Dennis, Eileen}

AC, AE, BC, BE, CA, CB, CD, DC, DE, EA, EB, ED 12

Assuming all members are eligible, but no one can hold more than one office, list and count the different ways the club could elect a president, a secretary, and a treasurer if the president must be a man and the other two must be women. (Carol & Erica are women, and the others are men.) N= {Aaron, Ben, Carol, Dennis, Erica} or, in abbreviated form, N= {A, B, C, D, E}

ACE, AEC, BCE, BEC, DCE, DEC 6

empirical rule

About 68% of all data values of a normal curve lie within 1 standard deviation of the mean (in both directions), about 95% within 2 standard deviations, and about 99.7% within 3 standard deviations.

Classify the triangle as acute, right, or obtuse and as equilateral, isosceles, or scalene.

Acute, equilateral

classify triangle all sides equal

Acute, equilateral

Classify the triangle as acute, right, or obtuse and as equilateral, isosceles, or scalene.

Acute; isosceles

acute triangle

All Angles Acute

equilateral triangle

All Sides Equal

R = {7, 8, . . . , 15, 16}

Although only 4 elements are listed, the ellipsis points indicate that there are other elements in the set. Counting them all, we find that there are 10 elements, so n(R) = 10

acute angle

An angle whose measure is between 0° and 90°

point

An exact location; a dot on a line: Usually represented by a Capital Letter

interior angles on same side of transversal

Angle measures add to 180°

alternate exterior angles

Angle measures are equal.

alternate interior angles

Angle measures are equal.

corresponding angles

Angle measures are equal.

straight angle

Angle that measures 180°

inscribed

Any angle inscribed in a circle has degree measure half of that of its intercepted arc

Given a group of students: Upper G = {Allen, Brenda, Chad, Dorothy, Eric} or G = {A, B, C, D, E} , list and count the different ways of choosing the following officers or representatives for student congress. Assume that no 1 can hold more than one office. 3 representatives, if 2 must be female and 1 must be male

BDA,BDC,BDE: 3

Consider the following counting problem. 8 women and 7 men are waiting in line at a movie theater. How many ways are there to arrange these 15 people amongst themselves such that the 8 women occupy the 1st 8 places and the 7 men the last 7 places? To solve this problem, which of the following rules would you use?

Both the permutations rule and the fundamental counting principle

space figure

Boxes are one kind of

Construct a tree diagram showing all possible results when three fair coins are tossed. Then list the ways of getting the following result. fewer than two heads

C TTT,HTT,THT,TTH

Determine whether the object is a permutation or a combination. a 10-digit telephone number (including area code)

Choose the correct answer below: This is neither a permutation nor a combination because repetition is allowed.

Determine whether the statement is true or false. The negation of "If it is winter, it is cold " is "If it is winter, it isn't cold."

Choose the correct answer below? False

Let T be a translation having magnitude four fifths inch to the right in a direction parallel to the bottom edge of the page. Let r Subscript m be a reflection about line m, and let Upper R Subscript p be a rotation about point P having magnitude 40 degrees clockwise. Perform the given transformation on point A of the figure to the right to obtain final image point A'.

Choose the figure below that shows the correct transformation of Upper T times Upper R Subscript p.

asymptote 12.5

Close but Never Touching When a curve approaches closer and closer to a line, without ever actually meeting it (as a normal curve approaches the horizontal axis)

Consider the selection of a 13 card bridge hand. Is this a combination, a permutation, or neither?

Combination

"A Prime"

Complement of A, contains all elements that are contained in "U" but are not contained in "A"

Consider only the smallest individual cubes and assume solid stacks (no gaps). Determine the number of cubes in the stack shown on the right that are not visible from the perspective shown.

Cubes not visible 10

binomial probability formula

Define the following quantities. n = the number of repeated trials p = the probability of success on any given trial q = 1−p=the probability of failure on any given trial a = the number of successes that occur

B= {1, 1, 2, 3, 2}

Do not count repeated elements more than once. Set B has only three distinct elements, so n(B) =

triangle size 12ft; 16ft whar is side (c) ft

Find the length of the third side of the right triangle. c=20 ft

For $3.98 you can get a salad, main course, and dessert at the cafeteria. If you have a choice of 4 different salads, 7 different main courses, and 5 different desserts, then how many different meals can you get for $3.98?

For the event of choosing a salad, the number of possible outcomes is 4. For the event of choosing a main course, the number of possible outcomes is 7. For the event of choosing a dessert, the number of possible outcomes is 5. Applying the fundamental counting principle you have 4 x 7 x 5=140.

Identify the set as finite or infinite. StartSet

INFINITE

antecedent - 102

Multiplication Rule of Probability (Event A & B)

If A and B are any two events, then P(A and B)=P(A)⋅P(B|A). If A and B are independent, then P(A and B)=P(A)⋅P(B).

Addition Rule of Probability (for the event "A and B)

If A and B are any two events, then the following holds. P(A or B)=P(A)+P(B)−P(A and B) If A and B are mutually exclusive, then the following holds. P(A or B)=P(A)+P(B)

Write the statement in the form 'if p, then q.' All natural numbers are real. choose statement that best rewrites the sentence in the specified form

If a number is natural, then it is always real

expected value of x

If a random variable x can have any of the values and the corresponding probabilities of these values occurring are P(x1),P(x2),P(x3),...,P(xn), then E(x)

percentile 12.4

If approximately n percent of the items in a distribution are less than the number x, then x is the nth percentile of the distribution, denoted Pn.

Let s represent 'she has a pet ocelot,' let t represent 'he trains tigers,' and let r represent 'they raise hamsters.' Express each compound statement in words. t -> ~r

If he trains tigers, then they do not raise hamsters

Write the negation of the statement. If it is not purple, then it is a cucumber .Choose the correct negation below:

If it is not purple and it is not a cucumber.

Let s represent "she paints for a living," let t represent "he fixes boats," and let r represent "they collect classics." Express the compound statement in words. What are the words that express the conditional statement

If she does not paint for a living, then he fixes boats.

complementary

If the sum of the measures of two acute angles is 90°

side-side-side

If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent.

angle-side-angle

If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent.

supplementary

If two angles have a sum of 180

side-angle-side

If two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent.

Rewrite the following statement using the if ... then connective. Rearrange the wording or add words as necessary. You can afford it if you see it on the television.

If you see it is on the television then you can afford it.

hypotenuse

In a right triangle, the side opposite the right angle (the longest side)

distinguishable arrangements

In counting arrangements of objects that contain look-alikes, the normal factorial formula must be modified to find the number of truly different arrangements. For example, the number of distinguishable arrangements of the letters of the word DAD is not 3!=6 3 factorial equals 6 but rather 3!2!=3. fraction 3 factorial , over 2 factorial end fraction . equals 3 . The listing below shows how the six total arrangements consist of just three groups of two, where the two in a given group look alike.

Non-Symmetric 12.2

In distributions skewed to the left (a), the data points start low and gradually go up. In distributions skewed to the right (b), data points go up quickly and then gradually go down. In bimodal distribution (c), data points form two peaks.

Use one of De Morgan's Laws to write the negation of the statement. It is June and there is no rain. What is the negation

It is not June and there is rain.

A lot is in the shape of a triangle. One side is 600ft longer than the shortest side, while the third side is 700ft longer than the shortest side. The perimeter of the lot is 4000ft. Find the lengths of the sides of the lot.

Lengths of the sides of the lit are: 900; 1500; 1600 ft.

properties of probability

Let E be an event within the sample space S. That is, E is a subset of S. Then the following properties hold. 1. 0≤P(E)≤1 The probability of an event is a number from 0 through 1, inclusive. 2. P(∅)=0 The probability of an impossible event is 0. 3. P(S)=1 The probability of a certain event is 1.

converting between probability and odds

Let E be an event. • If P(E)=ab, then the odds in favor of E are a to (b−a). • If the odds in favor of E are a to b, then P(E) = a / a+b.

glide reflection

Let rm r sub m be a reflection about line m, and let T be a translation having nonzero magnitude and a direction parallel to m. Then the composition of T and rm Here a reflection followed by a translation is the same as a translation followed by a reflection, so in this case

theoretical probability

Likewise, an established theoretical probability for an event enables us, by "deductive reasoning", to predict the proportion of times the event will occur in a series of repeated experiments. The prediction should be more accurate for larger numbers of repetitions.

Sets of Numbers

Natural numbers (or counting numbers); Whole numbers; Integers; Rational Numbers; Real Numbers; Irrational numbers;

A balanced die is rolled twice. Are the events "the sum of the two rolls is 8" and "the first roll comes up 3" independent ?

Decide whether or not the events are mutually exclusive. Being a male and being a nurse

In a city, 12% of the people drive Cadillacs and 21% of business executives drive Cadillacs. Are the events "person drives a Cadillac" and "person is an executive" independent?

In a city, 20% of the people drive Cadillacs and 20% of business executives drive Cadillacs. Are the events "person drives a Cadillac" and "person is an executive" independent?

In a city, 5% of the people drive Cadillacs and 30% of business executives drive Cadillacs. Are the events "person drives a Cadillac" and "person is an executive" independent?

scalene triangle

No Sides Equal

Write a negation for the following statement. Some shows are longer than this show.

No shows are longer than this show

coefficient of variation 12.3

Often this is a more meaningful measure than a straight measure of dispersion, especially when we are comparing distributions whose means are appreciably different.

obtuse triangle

One Obtuse Angle

right triangle

One Right Angle

Refer to the table below. Of the 36 possible outcomes, determine the number for which the sum (for both dice) is 12.

One can roll a sum of 12in _1__way(s).

Refer to the table below. Of the 36 possible outcomes, determine the number for which the sum (for both dice) is 3.

One can roll a sum of 3 in __2__ way(s).

empirical probability formula

P(E) = #of times event E occurred / # of times the experiment was performed If E is an event that may happen when an experiment is performed, then an empirical probability of event E is given by the following formula

Theoretical Probability Formula

P(E) = P (event) = # of favorable outcomes / total # of outcomes <<>>If all outcomes in a sample space S are equally likely, and E is an event within that sample space

triangle perimeter

P=a+b+c

point reflection

Point Q bisects the line segment from a point A to its image A', eh prime comma and for this reason this rotation is sometimes called

regular polygon

Polygons with all sides equal and all angles equal

On the 16 numbers in the product table, list the ones that belong in the category: 2, 3, 7, 9

Prime numbers are: 23,29,37,73,79,97

Label a segment whose endpoints are Upper Q and Upper R. Determine an appropriate name for a segment whose endpoints are Upper Q and Upper R. Choose the correct segment whose endpoints are Upper Q and Upper R Choose an appropriate name for a segment whose endpoints are Upper Q and Upper R.

Q(end-----end)R Q(end-----end)R

Classify the triangle Right angle triangle

Right, scalene

Classify the triangle as equilateral, isosceles, or scalene. Then classify the triangle as right, obtuse, or acute. Triangle with sides 63,49,21

Scalene; Obtuse

K = {3, 9, 27, 81

Set K contains 4 elements, so the cardinal number of set K is 4, and n (K) = 4

(M) = {0}

Set M contains only one element, 0, so n(M) = 1

How many ways can a teacher give 3 different prizes to 3 of her 25 students?

She an award the prizes __13,800__ ways.

How many ways can a teacher give 6 different prizes to six of her 21 students?

She can award the prizes in __39,070,080_ ways.

How many ways can a teacher give 5 different prizes to 5 of her 23 students?

She can award the prizes in __4,037,880___ ways.

Write the negation of the statement. No planes have three wheels.

Some planes have 3 wheels

conditional probability

Sometimes the probability of an event must be computed using the knowledge that some other event has happened (or is happening, or will happen—the timing is not important)

Testing the validity of an Argument with a Truth Table - 124

Step 1 Assign a letter to represent each component statement in the argument. Step 2 Express each premise and the conclusion symbolically. Step 3 Form the symbolic statement of the entire argument by writing the conjunction of all the premises as the antecedent of a conditional statement, and the conclusion of the argument as the consequent. Step 4 Complete the truth table for the conditional statement formed in Step 3. If it is a tautology, then the argument is valid; otherwise, it is invalid.

median 12.2

Step 1 Rank the items (that is, arrange them in numerical order from least to greatest). Step 2 If the number of items is odd, the median is the middle item in the list. Step 3 If the number of items is even, the median is the mean of the two middle items.

set equality (alternative definition)

Suppose A and B are sets: A = B is A B and B A are both true

theorems

The Greeks were the first to insist that all propositions about geometry be given rigorous proofs before being accepted

area

The amount of plane surface covered by a polygon measured in square units

Jessica's class schedule for next semester must consist of exactly one class from each of the four categories shown in the table. Category Number of Choices: Economics 6 Mathematics 3 Education 4 Sociology 4 All sections for the 3 most popular classes in Economics are full. The rest of the courses are available. Determine the number of different sets of classes Jessica can take.

The basic task is to design a schedule with a class from each category. There are four components to this task. The number of available Economics courses must be changed since 3 of the courses are full. The number of available Economics courses is 6-3=3 Now use the fundamental counting principle to multiply the number of options in each category. 3 x 3 x 4 x 4 =144 Jessica has 144 different sets of classes to choose from.

vertex

The common endpoint of the rays of the angle

Conditional Probability Formula

The conditional probability of B given A is calculated as P(B|A) = P(A∩B) / P(A) = P(A and B) / P(A)

Conditional Probability Formula

The conditional probability of B given A is calculated as P(B|A) = P(A∩B) / P(A) = P(A and B) / P(A). If A and B are mutually exclusive, then the following holds. P(A or B)=P(A)+P(B)

disjunction - 91-92

The disjunction p∨q p or q is false only if both component statements are false.

circumference

The distance around a circle is its circumference (rather than its perimeter)

magnitude

The distance between a point and its image under a translation The measure of angle ABA'

Many mathematical objects that are studied have dimensions that are whole numbers. For example, such solids as cubes and icosahedrons have dimension three. Squares, triangles, and many other planar figures are two-dimensional. Lines are one-dimensional, and points have dimension zero. Consider a square with side of length one, such as the one shown to the right. The size of a figure is calculated by counting up the number of replicas (small pieces) that make it up. Here, a replica is the original square with edges of length one. What is the least number of these squares that can be put together edge to edge to form a larger square? Square 1

The least number of squares is 4

mean (arithmetic mean) 12.2

The mean of a set of numbers is found by adding all the values in the set and dividing by the number of values 1) most common measure of central tendency "x bar" 2) The mean of n data items x1,x2,..., xn, x sub 1 , comma , x sub 2 , comma dot dot dot comma , x sub n , comma is calculated as follows. "x bar"=(∑x)/n 3) average

Pythagorean triple

The natural numbers 3, 4, and 5 form the Pythagorean triple (3, 4, 5) because they satisfy the equation of the Pythagorean theorem

Find the number of combinations (subsets) of 8 things taken 3 at a time.

The number is __5____ combinations

factorial formula for permutations

The number of permutations, or arrangements, of n distinct things taken r at a time, where r≤n, can be calculated as follows. nPr = n!/(n−r)!

A customer ordered 17 zingers. Zingers are placed in packages of 4, 3, or 1. In how many different ways can this order be filled?

The number of possible different ways is 17.

Determine the total number of proper subsets of the set of letters from the English alphabet {a, b, c, ..., l}.

The number of proper subsets is __4095_____.

Jason wants to dine at 5 different restaurants during a summer getaway. If 2 of 8 available restaurants serve seafood, find the number of ways that at least 1 of the selected restaurants will serve seafood given the condition that the order of selection is important.

The number of ways that at least one of the selected restaurants will serve seafood is __6000__

Jason wants to dine at 3 different restaurants during a summer getaway. If 2 of 6 available restaurants serve seafood, find the number of ways that at least 1 of the selected restaurants will serve seafood given the condition that the order of selection is important.

The number of ways that at least one of the selected restaurants will serve seafood is __96___.

Jason wants to dine at 5 different restaurants during a summer getaway. If 2 of 10 available restaurants serve seafood, find the number of ways that at least 1 of the selected restaurants will serve seafood given the condition that the order of selection is important.

The number of ways that at least one of the selected restaurants will serve seafood is ____23250______.

legs

The other two sides, which are perpendicular

center of rotation

The point of intersection of these two nonparallel lines

Conditional probability of B given A

The probability of event B, computed on the assumption that event A has happened, is called the conditional probability of B given A and is denoted P(B|A)

Probability of a Complement (For the Event "NOT E")

The probability that an event E will not occur is equal to 1 minus the probability that it will occur. P(not E)= 1 − P(E) P(E) + P (E') = 1 P(E) = 1 - P(E')

Decide whether the following is a statement or is not a statement. Greenville, California is on a mountain.

The sentence is a statement because it is either true or false

Match the following set with the appropriate description: {1, 3, 5, 7, 9}

The set of odd positive integers less than 10

Match the following set with the appropriate description. {2, 4, 6, 8, 10}

The set of the 5 least positive integer multiples of 2

Evaluate the expression. 6P3

The solution is 120

Evaluate the expression 9P2

The solution is 72

Evaluate 14! / 11!

The solution is _2184__ .

Refer to the groups of art labeled A, B, and C, and identify by letter the group or groups that are satisfied by the given statement involving a quantifier. No picture does not have a frame.

The statement is satisfied in group C.

Refer to the groups labeled A, B, and C, and identify by letter the group or groups that are satisified by the given statement involving a quantifier. At least one circle is not filled in with black.

The statement is satisfied in A and B

Symmetric 12.2

The three figures all show symmetric formations. In uniform distribution (a), the data points have the same vertical value. In binomial distribution (b), data points go up, hit a peak, and then go down. In bi-modal distribution (c), data points form two peaks.

random variable

The time spent on homework is an example of a random variable. It is "random" because we cannot predict which of its possible values will occur.

determine the number of squares (of any size) in the figure

The total number of squares is 30

Determine the number of triangles (of any size) in the figure.

The total number of triangles _20_

Determine the number of triangles (of any size) in the figure.

The total number of triangles is __16______.

Determine the number of triangles (of any size) in the figure.

The total number of triangles is ___12_____.

Give the number of rows in the truth table for the following compound statement.

The truth table consists of 128 rows

If the truth table for a certain compound statement has 16 rows, how many distinct component statements does it have?

The truth table has 4 distinct component statements.

consequent - 102

Then

Of the 16 numbers in the product table, list the ones that belong in the category: Prime Numbers: 2,3,7,9

There are 18 outcome where the sum is odd

Find the number of distinguishable arrangements of the letters of the word. QUINTILLION

There are _1,663,200___ distinguishable arrangements

A baseball team has 6 pitchers, who only pitch, and 16 other players, all of whom can play any position other than pitcher. For Saturday's game, the coach has not yet determined which 9 players to use nor what the batting order will be, except that the pitcher will bat last. How many different batting orders may occur?

There are _3,113,510,400_ different batting orders

How many 2-digit counting numbers are not multiples of 5?

There are _72____ 2-digit counting numbers which are not multiples of 5.

How many 2-digit counting numbers are not multiples of 10?

There are _81_ 2-digit counting numbers which are not multiples of 10.

Find the number of distinguishable arrangements of the letters of the word. MILLIARD

There are __10,080_ distinguishable arrangements.

A baseball team has 8 pitchers, who only pitch, and 9 other players, all of whom can play any position other than pitcher. For Saturday's game, the coach has not yet determined which 9 players to use nor what the batting order will be, except that the pitcher will bat last. How many different batting orders may occur?

There are __2,903,040____ different batting orders

Find the number of distinguishable arrangements of the letters of the word. TREDECILLION

There are __59,875,200___ distinguishable arrangements

A baseball team has 5 pitchers, who only pitch, and 14 other players, all of whom can play any position other than pitcher. For Saturday's game, the coach has not yet determined which 9 players to use nor what the batting order will be, except that the pitcher will bat last. How many different batting orders may occur?

There are __605,404,800__ different batting orders

collinear

Three points that lie on the same straight line

A multiple-choice test consist of 6 questions with each question having 3 possible answers.

To find the number of ways to mark the answers, use the Fundamental Counting Principle (or Multiplication Principle). Notice you cannot use permutations since repetitions are possible. Determine the number of possible outcomes for each event and multiply these together. For the event of choosing an answer for the first question, the number of possible outcomes is 3. For the event of choosing an answer for the second question, the number of possible outcomes is 3. Continuing in this manner, you can see that for each of the 6 events of choosing an answer, there are 3 possible outcomes. The number of ways to mark the answers is 3 x 3 x 3 x 3 x 3 x 3 = 729.

congruent triangles

Triangles that are both the same size and the same shape

Decide whether the following statement is true or false. "Given that p is true and ~q is false, the condition p -> q is true."

True

Determine whether the statement is true or false. Let A = {1, 3, 5, 7}, B= {5, 6, 7, 8}, C= {5, 8}, D= {2, 5, 8}, U = {1, 2, 3, 4, 5, 6, 7, 8} {6, 5, 8, 7} subset = B

True

isosceles triangle

Two Sides Equal

independent events

Two events A and B are independent events if knowledge about the occurrence of one of them has no effect on the probability of the other one occurring. A and B are independent if: P(B|A)=P(B), or equivalently, P(A|B)=P(A).

mutually exclusive events

Two events A and B are mutually exclusive events if they have no outcomes in common. Mutually exclusive events cannot occur simultaneously.

Let U = {1,2, 4, 5, a, b, c, d, e}. Find the complement of the set. P=@

subsets of a set

U = {1,2,3,4,5}, while A = {1,2,3} Set A is a subset of set B if every element of A is also an element of B

composition or product

We shall use the symbol rm r sub m to represent a reflection about line m, and let us use rn⋅rm r sub n , dot , r sub m to represent a reflection about line m followed by a reflection about line n. We call rn⋅rm

Draw a circuit representing the following statement. Simplify if possible. (~s^r) v s

What is a circuit representing the statement? Simplify the statement if possible s v r

Write a logical statement representing the circuit to the right. Simplify the circuit if possible.

What is the logical form for the given circuit? (~q^~p)vr Simplify the circuit. Choose the correct answer below? The circuit cannot be simplified further

Two angles are supplementary. One is 15 degrees more than four times the other. Find the measures of the angles. Supplementary angles are angles whose sum is 180degrees.

What is the measure of the smaller angle? 33 degrees (Simplify your answer.) What is the measure of the other angle? 147 degrees (Simplify your answer.)

fundamental counting principle

When a task consists of k separate parts and satisfies the uniformity criterion, if the first part can be done in n1 n sub 1 ways, the second part can then be done in n2 n sub 2 ways, and so on through the kth part, which can be done in nk n sub k ways, then the total number of ways to complete the task is given by the following product.

binomial probability formula

When n independent repeated trials occur, where p = probability of success and q = probability of failure with p and q (where q = 1 − p) remaining constant throughout all n trials, the probability of exactly x successes is calculated as follows. P (x) = nCxpxqn−x=n!x!(n−x)! pxqn−x

Write the following statement as an equivalent statement that does not use the if-then connective. Remember that p -> q is equivalent to ~p v q. If you scratch my back, I'll scratch yours.

You do not scratch my back or i will scratch yours

diameter

a chord that passes through the center

subset

a collection of some of the members; may be all members of the original set, or even none of them, or anywhere in between

conditional statement - 102

a compound statement that uses the connective if...then

range 12.3

a data set is a straightforward measure of dispersion (greatest value in the set) - (least value in the set)

statement - 84

a declarative sentence that is either true or false (not both simultaneously)

probability

a good measure of likelihood

tangent

a line that touches (intersects) the circle in only one point

set

a mathematical term for a group of objects

volume

a measure of capacity of a space figure measured in cubic units

skewed to the left 12.2

a non-symmetric distribution with a tail extending out to the left shaped like a J

vertical angles

a pair of angles that extend to form another angle vertical angles that have equal measures

rectangle

a parallelogram with a right angle (and consequently, four right angles)

rhombus

a parallelogram with all sides having equal length

trapezoid

a quadrilateral with one pair of parallel sides

parallelogram

a quadrilateral with two pairs of parallel sides 4-sided figure with both pairs of opposite sides parallel

square

a rectangle with all sides having equal length

chord

a segment whose endpoints both lie on the circle

polygon

a simple closed curve made up only of straight line segments.

convex

a simple closed figure that surrounds 2 points A and B inside the figure

normal curve 12.5

a symmetric, bell-shaped curve

tree diagram

a systematic way of listing all the subsets of a given set

central tendency 12.2

a traditional measurement of mode

Simpson's paradox 12.2

a trend appears in different groups of data but disappears when they are combined

Ethel's collection of eleven albums includes one jazz album. Ethel will choose five of her albums to play on a road trip. (Assume order is not important.) a) How many different sets of five albums could she choose? b) How many of these sets would not include the jazz album? c) How many of them would include the jazz album?

a) Ethel can choose from __462_ different sets of albums. b) There are _252__ sets that would not include the jazz album. c) There are __210___ sets that would include the jazz album.

Five men and five women have just six tickets in one row to the theater. In how many ways can they sit if the men and women are to alternate with either a man or a woman in the first seat?

a.) The first seat can be anyone. There are 10 choices of people for the first seat. b.) The second seat must be someone from the opposite sex of the person in the first seat. There are 5 choices for the second seat. c.) The third seat must be someone of the same sex as the first seat. There is one person sitting in the first seat already. There are 4 choices for the third seat. d.) The fourth seat must be the same sex as the second seat. There is one person sitting in the second seat already. There are 4 choices for the fourth seat. e.) The fifth seat must be someone of the same sex as the first and third seats. Two people of this sex are already seated. There are 3 choices for the fifth seat. f.) The sixth seat must be someone of the same sex as the second and fourth seats. Two people of this sex are already seated. There are 3 choices for the sixth seat. g.) Now find the product 10 x 5 x 4 x 4 x 3 x 3. 10 x 5 x 4 x 4 x 3 x 3 = 7,200 Therefore, five men and five women can sit down in 6 seats in 7,200 ways alternating sexes with either a man or a woman in the 1st seat.

Find the unknown lengths in the pair of similar triangles. (Triangles are not drawn to scale. Assume corresponding sides are in the same position within each triangle.)

a: 6 b: 21/2

Find the unknown lengths in the pair of similar triangles. (Triangles are not drawn to scale. Assume corresponding sides are in the same position within each triangle) 12x12x17 4xa4b

a=4 b=17/3 keep as fraction

Cathy's collection of twelve albums includes one rock album. Cathy will choose four of her albums to play on a road trip. (Assume order is not important.) a) How many different sets of four albums could she choose? b) How many of these sets would not include the rock album? c) How many of them would include the rock album?

a=495 b=330 c=165

Find all unknown angle measures in the pair of similar triangles. Note that the triangles are not drawn to scale. a= b=66 c=25 m= n= p=

a=89 b=66 c=25 m=66 n=89 p=25

Negation: Statement - 87

all do -- Some do not (Not all do) Some do -- None do (all do not)

truth table - 91

all possible combinations of truth values for the component statements

right angle

an angle that is 90°

invalid - 118

an argument that is not valid

fallacy - 118

an invalid argument arguments exhibiting illogical reasoning

Connective Symbol Type of Statement -

and ∧ Conjunction or ∨ Disjunction not ∼ Negation

obtuse angle

angles that measure more than 90° but less than 180°

perpendicular bisect

any chord of a circle passes though the center of the circle

experiment

any observation, or measurement of random phenomenon

normal distribution 12.5

any random variable whose graph has this characteristic shape (bell-shaped)

event

any subset of a sample space

premises - 117

assumptions, laws rules, widely held ideas, or observations

variance

average that results is itself a measure of dispersion

standard deviation 12.3 Pg 672

based on deviations from the mean of the data values >>> Let a sample of n numbers x1,x2,...,xn have mean x¯. Then the sample standard deviation, s, of the numbers is calculated as follows. s=∑(x−x¯¯)2n−1−−−−−−−−−−−√ The individual steps involved in this calculation follow. Step 1 Calculate x¯,the mean of the numbers. Step 2 Find the deviations from the mean. Step 3 Square each deviation. Step 4 Sum the squared deviations. Step 5 Divide the sum in Step 4 by n−1. Step 6 Take the square root of the quotient in Step 5.

Identify the curve as simple, closed, both, or neither. closed curve

both

rectangular parallelepiped

box

Sides a and b represent the two legs of a right triangle, and c represents the hypotenuse. Find the length of the unknown side. aequals 25 m, bequals 60 m

c= 65 m

Sides a and b represent the two legs of a right triangle, and c represents the hypotenuse. Find the length of the unknown side. a=7 b=24

c=25

simple curve

can be drawn without lifting the pencil from the paper and without passing through any point twice

one-part task

can be listed easily; such as tossing a coin

elements

can be numbers, words, objects, etc.

deterministic phenomenon

can be predicted exactly on the basis of obtainable information

standard normal curve

can be used to analyze any normal (or approximately normal) distribution a type of normal curve where the mean is 0 and the standard deviation is 1

random phenomenon

cannot be predicted exactly with obtainable information

set

collection or group of things; commonly designated using a list within braces, as we have been designating the club

odds

compares the number of favorable outcomes to the number of unfavorable outcomes >>against >>in favor >>>>if all outcomes in a sample space are equally likely, a of them are favorable to the event E, and the remaining b outcomes are unfavorable to E: * odds in favor of E are "a to b"* *odds against E are (b to a)*

tetrahedron

composed of four equilateral triangles, each three of which meet in a point.

octahedron

composed of groups of four regular (i.e., equilateral) triangles meeting at a point.

translation

composite transformation

fallacy of the converse - 124

conditional and converse are NOT equivalent

Construct a tree diagram showing all possible results when 3 fair coins are tossed. Then list the ways of getting the following result. @ least 2 tails

construct a tree diagram. Choose the correct diagram below. A Select the correct choice below that lists the appropriate branches. TTH,THT, HTT, TTT

Decide whether the figure is convex or not convex. Choose the correct answer below.

convex

Natural numbers

counting numbers {1,2,3,4,....}

skew lines

do not lie in a common plane, so they are neither parallel nor intersecting

z-score 12.4

each individual item in a sample can be assigned >> the 1st measure of position that we consider >>>If "x" is a data item in a sample with mean "x bar" and standard deviation "s", then the z-score of x is calculated as follows:

Complete the blank with either is an (element of)or is (not an element) of to make the statement true. {6} _______ {{3}, {4}, {5}, {6}, {7}}

element

size transformation

every point of the image semicircle, such as C', c prime comma was obtained by drawing a line through M and C, and then locating C' c prime such that |MC'|=2|MC|

empirically

experimentally

Decide whether the statement is true or false. If a quadrilateral is a parallelogram, then it must be a rhombus.

false

Decide whether the statement is true or false. Is a quadrilateral is a parallelogram, then it must be a rhombus

false

plane figure

figures that can be drawn completely in the plane of a sheet of paper.

counting

finding the number of objects, of some certain type, that exist

center

fixed point in the middle of a circle

complement of a set

for any set "A" within the universal set "U", the complement of "A", written A', is the set of elements of U that are not elements of "A" A' = {x|x U and x A

Chebyshev's theorem 12.3

for any set of numbers, regardless of hoe they are distributed, the fraction of them that lie within "k" standard deviations of ther mean (where k is greater than 1) is at least ( 1 - 1/k squared)

tree diagram

for tasks that have more than 2 parts is not easy to analyze with a product table

intersecting planes

form a straight line, the one and only line they have in common

dodecahedron

formed by groups of three regular pentagons

fair game

game in which the expected net winnings are zero

closed curve

has its starting and ending points the same and is drawn without lifting the pencil from the paper

prism

have two faces in parallel planes; these faces are congruent polygons

Use a tree diagram showing all possible results when 4 fair coins are tossed. Then list the ways of getting the indicated result. at least 2 tails

hhtt, htht, htth, httt, thht, thth, thtt, tthh, ttht, ttth, tttt

conditional statement

if I go, then you stay

contrapositive - 111

if the antecedent and the consequent are both interchanged and negated

valid - 118

if the truth of the premises are true forces the conclusion to be true

equivalent statements - 102

if they have the same truth value in every possible situation. The columns of the two truth tables that were the last to be completed will be the same for equivalent statements.

simulation

imitation an important area within probability theory

order

in all cases decide whether order is important, because that determines whether to use permutations or combinations

conditional connective

in no way implies a cause-and-effect relationship

universal set

in set theory the universe of discourse typically designated by the letter U

universe of discourse

includes all things under discussion at a given time

line segment

includes both endpoints and is named by its endpoints

ray

including an initial point with a half-line

existential qualifiers - 86

indicate at least one member: at least one

argument - 117

inductive reasoning we observe patterns to solve problems; > made up of premises and conclusion

converse - 111

interchanging the antecedent and the consequent

fallacy of the inverse - 125

invalid argument

transformational geometry

investigates how one geometric figure can be transformed into another

box plot (box-and-whisker plot) 12.4

involves the median (a measure of central tendency), the range (a measure of dispersion), and the first and third quartiles (measures of position), all incorporated into a simple visual display. >> consists of a rectangular box positioned above a numerical scale, extending from Q1 to Q3, with the value of Q2 (the median) indicated within the box, and with "whiskers" (line segments) extending to the left and right from the box out to the minimum and maximum data items

probability of any event

is a number from 0 through 1, inclusive

Decide whether the following statement is compound. Joan's mother likes to wrap gifts and she gives many to friends

is the statement a compound statement? Yes

finite set

is when the cardinal number of a set is a particular whole number (0 or a counting number)

intersecting lines

lie in the same plane and cross each other

parallel line

lie in the same plane and never meet, no matter how far they are extended.

transformation

line m is perpendicular to the line segment AA' eh eh prime and bisects this line segment

listing method

listing the elements, separated by commas, in squiggly brackets Example: {a, e, i, o, u}

Which of the lines in the picture are lines of symmetry of the given figure?

m and o

which of the lines in the picture are lines of symmetry of the given figure?

m and o

pyramid

made of triangular sides and a polygonal base

icosahedron

made up of groups of five regular triangles

compound statement - 84

made up of two or more component statements joined by connectives (not, and, or, if ... then).

theoretically

mathematically

measures of position 12.4

measure an items position within the data set

uniformity criterion

multi-part task; if the number of choices for any particular part is the same no matter which choices were selected for previous parts

Find n(A) for the set. A = {x | x is a month in the year}

n(A) = 12

Find n(A) for the set.:

n(A)=29

inverse - 111

negating both the antecedent and the consequent the contrapositive of the converse

Consider determining how many possible phone numbers are within a particular area code (repeated numbers allowed). Is this a combination, a permutation, or neither? A. Neither This is the correct answer. B. Combination C. Permutation

neither

parallel planes

never meet, no matter how far they extend

skewed to the right 12.2

non-symmetric distribution with a tail extending to the left

Decide whether the statement is compound. My backpack is red. Choose the correct answer below.

not compound

negation - 85-87

of a true statement has the opposite truth value of that statement in all cases: *true is false and false is true* >must have the opposite truth value from the original >>the negation if a statement is simply (the statement itself)

perimeter

of any polygon is the sum of the measures of the line segments that form its sides. Perimeter is measured in linear units.

truth values of conjunction - 91

of component statements are used to find truth values of compound statements

Refer to the table below. Of the 36 possible outcomes, determine the number for which the sum (for both dice) is 5.

one can roll a sum of 5 in "4" ways

half-line

one on each side of the point

hexahedron

or cube, is composed of six squares, each three of which meet at a point

biconditional - 114

p if and only if q ( p iff q) > is true when both component statements have the same truth value > is false when they have different truth values

geometric construction

page 63

similar triangles

pairs of triangles that are exactly the same shape but not necessarily the same size.

invariant point

point A and its image, A', eh prime , comma under a certain transformation are the same point

secant

point of tangency. PQ←→ modified p q with left right arrow above which intersects the circle in two points

outcomes

possible results of the experiment

possible sequences

possibly outcomes

conclusion - 117

premise and premise =

line

properties of no thickness and no width & extend indefinitely in 2 directions: may be represented by 2 Capital Letters representing points on that lie or line may also be represented by usually lower case *Cursive l* *subscript is sometimes used*

inclusive disjunction - 92

p∨q means " p or q" or both

palindromic

read the same forward and backward

line of symmetry

reflection images about the lines of reflection

n(A); which is read as "n of A"

represents the "cardinal number" of "set A"

symbol ∼ - 86

represents the connective not

simulation methods

require huge numbers of random digits, so computers are used to produce them.

classify triangle as acute, right, or obtuse and as equilateral, isosceles, scalene

right; scalene

radius

segments whose endpoints are in the center and a point on the circle

sample space

set of all possible outcomes the set of all possible outcomes of a probability experiment

Pamela's computer printer allows for optional settings with a panel of 4 on-off switches in a row. How many different settings can she select if there are no restraints

she can select 16 different settings

Pamela's computer printer allows for optional settings with a panel of 3 on-off switches in a row. How many different settings can she select if there are no restrictions on the switches?

she can select 32 different settings

Pamela's computer printer allows for optional settings with a panel of 6 on-off switches in a row. How many different settings can she select if at least one switch must be on?

she can select 63 different settings

Pamela's computer printer allows for optional settings with a panel of 4 on-off switches in a row. How many different settings can she select if at least one switch must be on?

she can select __15__ different settings

Identify the curve as simple, closed, both, or neither.

simple

Identify the curve as simple, closed, both, or neither. shape looks like W

simple

Decide whether the following is a statement or is not a statement. 2 + 6 = 8 and 5 + 6 not equals 5

statement

disjunction syllogism - 125

statement is a tautology and the argument is valid

Construct a truth table for the given statement. Identify whether the statement is a tautology. (p v q) -> (q v p) Construct a truth table.

statement is a tautology, since it is true for all combinations of truth values of the components

component statements - 84

statements making up a compound statement

Decide whether subset equals, is a proper subset of, both, or neither can be placed in the blank to make a true statement. {3,4,5} _______ {3, 4, 5}

subset =

Find the surface area and the volume of the cylinder. r=11 cm h=9 cm

surface area = 1381.60 cm squared volume = 3419.46 cm cubed

Find the surface area and the volume of the cylinder. Use 3.14 for pi .radius: 10 Height: 4

surface area: 879.20 volume: 1256

euler diagram - 118

technique to check the validity:: Premise; Premise = conclusion

Find the number of combinations (subsets) of 10 things taken 6 at a time.

the answer is 210 combinations

rotation

the composition of two reflections about nonparallel lines

Suppose carpet for a 10 ft by 8 ft room costs $300. Find the cost to carpet a room 40 ft by 32 ft.

the cost to carpet the room is "$4800"

Venn diagram

the entire region bounded by the rectangle represents the universal set U, and the portion bounded by the circle represents set A

polyhedra

the faces of which are made only of polygons

A search plane carries radar equipment that can detect metal objects (like submarine periscopes or plane wreckage) on the ocean surface up to 18.5 miles away. If the plane completes a circular flight pattern of 785 miles in circumference, how much area will it search? (Use 3.14 as an approximation for pi.) Three concentric circles are shown over the Indian Ocean. A plane lies between the inner and outer most circles on the right side. The circle in between the other two contains a dashed border and denotes the planes flight pattern.

the flight pattern has a total search area of approximately (29,000 square miles)

modus tollens - 125

the law of contraposition or indirect reasoning

reasoning by transitivity - 126

the law of hypothetical syllogism

symmetry in data sets 12.2

the most useful way to analyze a data set often depends on the distribution

cardinal number (cardinality)

the number of elements in a set: n(A); which is read as "n of A"

Determine the total number of proper subsets of the set of letters from the English alphabet {a, b, c, ..., h}.

the number of proper subsets is __255___.

number of proper subsets

the number of proper subsets of a set with _____ elements

number of subsets

the number of subsets of a set with _____ elements

(n) factorial

the quantity of n formula is defined as: n! = n * (n - 1) * (n - 2) * .... * 2 * 1

sides

the rays of an angle

reflection preserves collinearity

the reflection image of a line is also a line the reflection image of a line is also a line. We express this by saying that reflection preserves collinearity.

Match the following set with the appropriate description. {..., -4, -3, -2, -1}

the set of all negative integers

dispersion (spread) of data 12.3

the spread of information

tautology - 105

the statement is always true no matter what the truth value of the components

Use truth tables to decide whether the pair of statements is equivalent. q->p; ~q^p

the statements are not equivalent. when both p and q are true, the statements have different truth values

cumulative frequency 12.2

the sum of the frequencies for that class and all previous classes

Let p represent the statement "I get an Upper A" and q represent the statement "I graduate". Convert the following compound statement into symbols. Upper I get an A or I do not graduate.

the symbolic form is p v ~q

Let u represent the statement "Upper I eat bananas" and t represent the statement "Upper I work hard". Convert the following compound statement into symbols. Neither Upper I eat bananas nor Upper I work hard

the symbolic form is ~(uvt)

quartile 12.4

the three values ( Q1,Q2, q sub 1 , comma . q sub 2 , comma and Q3 q sub 3 ) that divide a data set into four (approximately) equal-sized parts >For any set of data (ranked in order from least to greatest): 1) The second quartile, Q2, is just the median, the middle item when the number of items is odd, or the mean of the two middle items when the number of items is even. 2) The first quartile, Q1 is the median of all items below Q2. 3) The third quartile, Q3 is the median of all items above Q2.

angle

the union of two rays that have a common endpoint

Venn diagrams

the universal set is represented by a rectangle, and other sets of interest within the universal set are depicted by circular regions

mode 12.2

the value that occurs most often

Let p represent the statement "The vest is tan " and let q represent the statement "The scarf is brown." Translate the symbolic compound statement ~p ^ q into words.

the vest is tan and the scarf is brown

weighted mean 12.2

the weighted mean of a group of (weighted) items is the sum of all products of items times weighting factors, divided by the sum of all weighting factors

sphere

three-dimensional analogue of a circle

conjunction p^q - 91

to be true, both p and q must be true

protractor

tool used to measure angles

surface area

total area that would be covered if the space figure were "peeled" and the peel laid flat

isometry

transformations in which the image of a figure has the same size and shape as the original figure

Decide whether the statement is true or false. If a triangle is equilateral (all sides the same), then it cannot be obtuse. False True Your answer is correct.

true

Let p represent a true statement and let q represent a false statement. Find the truth value of the given compound statement. p v q

true

there is no angle that is the complement of an obtuse angle

true

If pv(q^~q) is true, what must be the truth value of p?

truth value or p? true

semicircle

two parts of equal size of a circle

right circular cylinder

typical tin can is example

two-part task

use product table

quantifiers - 86

used to indicate how many members in a group being considered exhibit a particular property or characteristic:

symbolic logic - 84

uses letters to represent statements, and symbols for words such as and, or, not.

skewness coefficient 12.3

uses to measure the degree of skewness involves both central tendency and dispersion and is calculated

set-builder notation

uses variables to define elements Example: { x | x is a vowel}

continuous random variable

variable whose values are not restricted

product table

vertical and horizontal table of values

In order to determine the amount of liquid a spherical tank will hold, would you need to use volume or surface area? Choose the correct answer below.

volume

find the surface area and volume of the figure

volume 41.87 surface area 76.6

Find (a) the volume and (b) the surface area of the sphere. Use 3.14 as an approximation for pi .contact and eyeball shape

volume 57876.48 surface area 7234.56

Find the volume and the surface area of the sphere. sphere diameter = 28ft.

volume = 11488.21 ft cubed surface area = 2461.76 ft. squared

Find the surface area and volume of the figure: height = 10 in radius = 2 in

volume = 41.87 in cubed surface area = 76.6 in squared

Find (a) the volume and (b) the surface area of the figure. l; 3 h; 2.3 w; 5

volume: 34.5 surface area: 66.80

find volume and surface area of the figure "rectangle" 6 x 4 x 2.6

volume= 62.4 surface area = 100

perpendicular lines

when 2 lines inter sect to form right angles

transversal line

when a line intersects 2 parallel lines

binomial Probability distribution

when outcomes of an experiment are divided into just 2 categories, success and failure, the associated probabilities are "binomial" Bernoulli trials after James Bernoulli

infinite set

whenever a set is so large that its cardinal number is not found among the whole numbers

written method

writing out explanation of the elements Example: "the set of vowels in the English alphabet"

one degree

written 1°, comma is defined to be 1/360 of a complete rotation

The box has a volume of 1575. . Find x. width= 14 length= 15 height= x

x = 7.5

The two triangles below are similar. Find the unknown side lengths.

x=4.9 y= 3.6

Decide whether or not the events are mutually exclusive. Events A and B defined as follows A card is drawn at random from a deck of cards. Event A is that the card obtained is a face card and B is the event that the card obtained is a two.

yes

List all the elements of the following set. Use set notation and the listing method to describe the set. {x|x is an odd integer between -8 and 5

{-7, -5, -3, -1, 1, 3}

List the elements in the set. {x | x is a negative multiple of 8}

{-8, -16, -24,...}

Integers

{..., -3, 2, 1, 0, 1, 2, 3, ... }

List all the elements of the following set. Use set notation and the listing method to describe the set. The set of all whole numbers not greater than 1

{0, 1}

Whole numbers

{0,1,2,3,4,.....}

List all the elements of the following set. Use set notation and the listing method to describe the set. The set of all counting numbers less than or equal to 6

{1, 2, 3, 4, 5, 6}

List all the elements of the following set. Use set notation and the listing method to describe the set. {x |x| is an odd whole number less than 11}

{1, 3, 5, 7, 9}

Let = {1, 2, 4, 5, a, b, c, d, e}. Find the complement of the set. Upper W = {1, 5, e, d, a}

{2, 4, b, c}

List all the elements of the following set. Use set notation and the listing method to describe the set. {6,7, 8, ..., 14}

{6, 7, 8, 9, 10, 11, 12, 13, 14}

List all the elements of the following set. Use set notation and the listing method to describe the set. The set of all whole numbers greater than 6 ad less than 15.

{7, 8, 9, 10, 11, 12, 13, 14}

A committee is to be formed. Possible candidates for the committee are Eric, Frances, Greg, and Jose. Denoting these four people by e, f, g, j, list all possible committees of two people (ie list all possible subsets of size two).

{e,f}, (e,g}, {e,j}, {f,g}, {f,j}, {g,j}

Rational numbers

{p/q | p and q are integerds and q~0}

Real Numbers

{x|x is a number that can be expressed as a decimal}

Irrational numbers

{x|x is a real number and x cannot be expressed as a quotient o integers}

Write the set in set-builder notation. {negative 5, negative 4, negative 3, negative 2, ...}

{x|x is an integer greater than -6}

If I do not eat bananas, then I do not get an A. Symbolic form:

~p -> ~q

The 69 students in a classical music lecture class were polled, with the results that 38 like Wolfgang Amadeus Mozart, 37 like Ludwig von Beethoven, 33 like Franz Joseph Haydn, 14 like Mozart and Beethoven, 22 like Mozart and Haydn, 14 like Beethoven and Haydn, and 8 like all three composers. Use a Venn diagram to complete parts (a) through (f) below. Complete the Venn diagram. Let M be the set of students that like Mozart, B be the set of students that like Beethoven, and H be the set of students that like Haydn. M=10 MB=6 B=17 MBH=8 MH=14 BH=6 H=5 U=3

(a) How many of these students like exactly two of these composers? 26 student(s) (Simplify your answer.) (b) How many of these students like exactly one of these composers? 32 student(s) (Simplify your answer.) (c) How many of these students like none of these composers? 3 student(s) (Simplify your answer.) (d) How many of these students like Mozart, but neither Beethoven nor Haydn? 10 student(s) (Simplify your answer.) (e) How many of these students like Haydn and exactly one of the other two? 20 student(s) (Simplify your answer.) (f) How many of these students like no more than two of these composers? 61 student(s) (Simplify your answer.)

Find (a) the volume and (b) the surface area of the figure. A rectangular box has length 9 inches, width 6 inches, and height 2.7 inches.

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