Math Glossary

Inductive reasoning:  the process of reasoning at a general conclusion or conjecture based on the observation of specific examples.

Deductive reasoning:  the process of reasoning that arrives at a specific conclusion based on previously accepted general statements.

Set:  a well-defined collection of objects.

Variable:  a symbol (usually a letter) that can represent different elements of a set.

Null set:  a set with no elements, sometimes called an empty set.  The symbols used to represent the null set are { } or a circle with a slash.

Two sets, A and B, are equal (written A = B) if they have exactly the same members or elements.

Two sets have a one-to-one correspondence if and only if it is possible to pair the elements of one set with the elements of the other set in such a way that for each element in the first set there exists one and only one element in the second set.

If every element of set A is also an element of set B, then set A is called subset of set B.  The symbol (c with a line under it) is used to designate a subset, and the relationship is written A (c with a line under it) B.

If a subset of a given set is not equal to the original set, then the subset is called a proper subset of the original set.  The symbol (c) is used to indicate a proper subset.

The union of two sets A and B (symbolized by A u B) consists of the elements of set A or set B, or both sets.

The intersection of two sets, A and B, symbolized by A n B, is the set of elements that are common to both sets.

The complement of set A, denoted by (A with a line over it) is the set of elements contained in the universal set that are not contained in A.

Infinitive set:  a set that can be placed in a one-to-one correspondence with a proper subset of itself.

Statement:  a sentence that can be determined to be true  or false but not both at the same time.

Connectives:  are "and," "or," "if. . .then," and "if and only if."

When a compound statement is always true, it is called a tautology.  When a compound statement is always false, it is called a self-contradiction.

Two compound statements are logically equivalent if and only if they have the same truth table values.  The symboy for logically equivalent statements is (3 parallel horizontal lines).

Natural numbers:  (N) consists of numbers 1, 2, 3, . . .

Prime number: a natural number that has only two factors, one and itself.

Composite number:  a natural number greater than 1 that has three or more factors.

Fundamental  theorem of arithmetic:  states that every composite number can be expressed as a product of prime numbers in only one way.  The order of the factors is disregarded.

Greatest common factor (divisor) (GCF):  of two or more numbers is the largest number that is a factor or divisor of all of the numbers.

Least common multiple (LCM):  of two or more numbers is the smallest number that is divisible by all the numbers.

Whole numbers (W):  defined as 0, 1, 2, 3, 4, 5, . . .

Integers (Z):  defined as . . . . -3, -2, -1, 0, 1, 2, 3, . . . .

Any number that can be written as a fraction, a/b, where a and b are integers and b (= with a slash through it) 0 is called a rational number.  The integer a is called the numerator of the fraction, and the integer b is called the denominator of the fraction.

Proper fraction:  has a numerator whose absolute value is less than the absolute value of the denominator.

Improper fraction:  has a numerator whose absolute value is greater than or equal to the absolute value of the denominator.

Mixed number:  consists of a whole number and a fraction.

A set of numbers is said to be dense if for any two given numbers in the set, there exists a third number in the set that  lies between the two given numbers.

The set of irrational numbers (I) consists of the numbers that can be written as non-terminating and nonrepeating decimals.

The set of real numbers (R) consist of the union of the set of rational numbers and the set of irrational numbers.

The closure property of addition:  For any two real numbers a and b, the sum a + b will be a real number.

The closure property of multiplication:  For any two real numbers a and b, the product a . b will be a real number.

The communtative property of addition:  For any real numbers a b, a + b = b + a.

The communtative property of multiplication:  For any real numbers a and b, a x b = b x a.

The associative property of addtion:  For any real number a, b, and c, (a + b) + c = a + (b + c).

The associative of multiplication:  For any real numbers a, b, and c, (a x b) x c = a x (b x c).

The identity property for addition:  For any real number a, there exists a real number zero called the identity for addition such that 0 + a = a, and a + 0 = a.  Zero is called the identity for addition.

The identity property for multiplication:  For any real number a, there exists a real number 1 called the identity for multiplication such that 1 x a = a and a x 1 = a.  The number 1 is called the identity for multiplication.

Inverse property for addition:  For any real number a, there exists a real number -a such that a + (-a) = 0 and -a + a = 0.  -a is said to be the additive inverse or opposite of a.

Inverse property for multiplication:  For any real number a, except a, except 0, there exists a real number 1/a such that a x 1/a = 1 and 1/a x a = 1.

The distributive property of multiplication over addition:  For any real numbers a, b, and c, a x (b + c) = a x b + a x c.

For any positive integer n, a(n power) = a x a x a x a x . . . ., a/n factors, where a is called the base and n is called the exponent.

When an exponent is negative, it is defined as follows:  For any positive integer n, a (-n power) = 1/a(n power).

For any number a, a (0 power) = 1.

A number expressed in scientific notation is written as a product of a number n, where n is 1 (< with a line under it) n < 10, and some power of 10.

A sequence of numbers is a list of numbers that are related to each other by a specific rule.  Each number in the sequence is called a term of the sequence.

A geometric sequence is a sequence of terms in which each term after the first term is obtained by multiplying the preceding term by a nonzero number.  This number is called the common ratio.

The nth term of a geometric sequence is a (sub n) = a (sub 1) r (n-1 power), where a (sub 1) is the first term and r is the common ration.

A numeration system consists of a set of symbols and various rules for combining the symbols to represent numbers.

A mathematical system is called a group if it has these properties:

     1.  The set of elements is closed for the binary operation.

     2.  There exists an identity element for the set.

     3.  Any three elements in the set are associative for the binary

          operation.

     4.  Every element has an inverse.

A variable, usually a letter, can represent different numerical values.

An algebraic expression consists of any meaningful combination of variables, numbers, operation symbols, and grouping symbols.

The distributive property of multiplication over addition states that, for any real numbers a, b, and c, a(b + c) = ab + ac.

An equation is a statement of equality of two algebraic expressions.

An open equation contains at least one variable.

The addition property of equality states that the same real numbers on algebraic expressions can be added to both sides of an equation without changing the solution set for the equation:  i. e., if a = b, then a + c = b + c.

The subtraction property of equality states that the same real number of algebraic expression can be subtracted from both sides of an equation without changing the solution set for the equation; i. e., if a = b, then a - c = b - c.

The multiplication property of equality states that the same nonzero real number can be multiplied to both sides of the equation without changing the solution set for the equation; i. e., if a = b and c (= with a slash through it) 0, then ac = bc.

The division property of equality states that both sides of an equation can be divided by the same nonzero real number without changing the solution set of the equation; i. e., if a = b and c (= with a slash through it) 0, then a/c = b/c.

Procedure for Solving Equations

Step 1:  Remove parentheses.

Step 2:  Combine like terms on each side of the equation.

Step 3:  Get the variables on one side of the equation and the numbers on

             the other side of the equation by using the addition and/or

             subtraction properties of equality.

Step 4:  Combine like terms.

Step 5:  Use the multiplication or division property of equality to solve for

             the variable.

Procedure for Solving Word Problems Using Equations

Step 1:  Read the problem carefully.

Step 2:  Let x represent an unknown quantity.

Step 3:  Write the equation based on the information given in the problem.

Step 4:  Solve the equation x.

Step 5:  Check the solution.

To solve a linear inequality, proceed as if you were solving a linear equation except that when multiplying or dividing by a negative number, you must reverse the inequality sign.

A ratio is a comparison of two quantities using division.

For two nonzero numbers, a and b, the ration of a to b is written as a:b (read a to b) or a/b.

A proportion is a statement of equality of tow rations.

Procedure for Solving Word Problems Using Proportions

Step 1:  Read the problem and find the ration statement.

Step 2:  Write the ratio using a fraction.

Step 3:  Set up the proportion using x for the unknown number.

Step 4:  Solve the proportion for x.

y s said to vary directly with x if there is some nonzero constant k such that y = kx.

y is said to very inversely with x if there is some nonzero constant k such that y = k/x.

The standard form of a quadratic equation is ax(2) + bx + c = 0, where a, b, and c are real numbers a (= with a slash through it) 0.

Rule of Signs for Factoring Trinomials

If the sign of the third term (i. e., the constant term) of the trinomial is positive, then the signs of its factors are both positive if the sign of the second term (i. e., the x term) is positive, or both negative if the sign of the second term is negative.

If the sign of the third term of the trinomial is negative, then the sign of one of its factors will be positive and the sign of the other factor will be negative.

Procedure for Solving Quadratic Equations by Factoring

Step 1:  Write the quadratic equation in standard form.

Step 2:  Factor the left side.

Step 3:  Set both factors equal to zero.

Step 4:  Solve each equation for x.

The formula x = -b +/- (the square root of) b(2) - 4ac/2a, is called the quadratic formula and can be used to solve any quadratic equation written in standard form, ax(2) + bx + c = 0, a (= with a slash through it) 0.

Finding Intercepts

To find the x-intercept, substitute 0 for y and solve the equation for x.

To find the y-intercept, substitute 0 for x and solve the equation for y.

The slope of a line (designated by m) is m = y(sub 2) - y(sub 1)/x(sub 2 - sub 1), where (x(sub 1), y(sub 1)) and (x(sub 2), y(sub 2)) are two points on the line.

The slope-intercept form for an equation in two variables is y = mx + b, where m is the slope and (0, b) is the point where the line crosses the y axis.

A system of two linear equation in two variables can be represented as

a(sub 1)x + b(sub 1)y = c(sub 1)

a(sub 2)x + b(sub 2)y = c(sub 2)

Procedure for Solving a System of Equations Graphically

Step 1:  Draw the graphs of the equations on the same Cartesian plane.

Step 2:  Find the point or points of intersection of the two lines if they

             exist.

Procedure for Solving a System of Equations by Substitution

Step 1:  Select one equation and solve it for one variable (either x or y) in

             terms of the other variable.

Step 2:  Substitute the expression containing the other variable that you

             found step 1 into the other equation.

Step 3:  Solve the equation for the unknown (it now has only one variables).

Step 4:  Select one of the original equations, substitute the value found in

             step 3 for the variable, and solve it for the value of the other

             variable.

Procedure for Solving a System of Equation Using the Addition/Subtraction (Elimination) Method

Step 1:   If necessary, write both equations in the form ax + by = c.

Step 2:  Multiply one or both equations by numbers so that the absolute

             values or either the coefficients of the x terms or the y terms are

             alike.

Step 3:  Eliminate one of the variables by adding the equations if the signs

             of the coefficients of the variable are different.  Subtract the

             equations if the signs of the coefficients of the variables are the

             same.

Step 4:  Solve the resultant equation for the remaining variable.

Step 5:  Select one equation from the original two equations, substitute the

             value of the variable found in step 4, and solve for the other

             variable.

A half plane is the set of points on the Cartesian plane that are on one side of a line.

Procedure for Using Linear Programming

Step 1:  Write the objective function.

Step 2:  Write the constraints.

Step 3:  Graph the constraints.

Step 4:  Find the vertices of the polygonal region.

Step 5:  Substitute the coordinates o f the vertices into the objective

             function and find the maximum or minimum value.

(Note:  The solutions will not always be integers.)

A relation is a set of ordered pairs of elements.

A relation is a function if for each element in the domain there is a unique in the range.

The Vertical Line Test for Functions

If no vertical line can intersect the graph of a relation of more than one point, then the relation is a function.

Procedure for Graphing a Linear Function

Step 1:  Select at least 3 values for x.  (Only two values are necessary.  The

             third is used as a check.)

Step 2:  Substitute them in the function and find the corresponding values

             for f(x).
Step 3:  Plot the points (x, f(x)) on the Cartesian plane using the y axis as

             the f(x) axis.

Step 4:  Draw a line through the points.

An equation of the form f(x) = ax(2) + bx + c, where a, b, and c are real numbers and a (= with a slash through it) 0, is called a quadratic function.  This equation can also be written as y = ax(2) + bx + c.  The graph of a quadratice function is called a parabola.

Procedure for Graphing the Quadratic Function y = ax(2) + bx + c

Step 1:  Identify a, b, and c, then find the vertex using x = -b/2a to get the

             x coordinate.  Then substitute this value in the equation y = ax(2) +

             bx + c to get the y cooordinate.

Step 2:  Find the x-intercepts by substituting x = 0 in the equation and

             solving for y or use (0, c).

Step 3:  Find the x-intercepts by substituting 0 for y and solving the

             equation for x, either by factoring or by using the quadratic

             formula.

Step 4:  Find several other points in order to determine the shape.

Step 5:  Determine whether the parabola opens upward (a > 0) or downward

             (a < 0).  Plot the points and draw a smooth curve through the points.

An exponential function has the form f(x) = a(x power), where a and x are real numbers such that a > 0 (= with a slash through it) 1.

Finding Intercepts

To find the x intercept, substitute 0 for y and solve the equation for x.

To find the y intercept, substitute 0 for x and solve the equation for y.

The slope of a line (designed by m) is m = y(2)-y(1)/x(2)-x(1), where (x(2), y(2)) are two points on the line.

The slope-intercept form for an equation in two variables is y = mx + b, where m is the slope and (0, b) is the point where the line crosses the y axis.

A system of two linear equations in two variables can be represented as

a(1)x + b(1)y = c(1)

a(2)x + b(2)7 = c(2)

Procedure for Solving a System of Equations Graphically

Step 1:  Draw the graphs of the equations on the same Cartesian plane.

Step 2:  Find the point or points of intersection of the two lines if they exist.

Procedure for Solving a System of Equations by Substitution

Step 1:  Select one equation and solve it for one variable (either x or y) in

             terms of the other variable.

Step 2:  Substitute the expression containing the other variable that you

             found in step 1 into the other equation.

Step 3:  Solve the equation for the unknown (it now has only one variable).

Step 4:  Select one of the original equations, substitute the value found in step                 

             3 for the variable, and solve it for the value of the other variable.

Procedure for Solving a System of Equations Using the Addition/Subtraction (Elimination) Method

Step 1:  If necessary, write both equations in the form ax + by = c.

Step 2:  Multiply one or both equations by numbers so that the absolute 

             values st either coefficients of the x terms  or the y terms are alike.

Step 3:  Eliminate one of the variables by adding the equations if the signs of

             the coefficients of the variable are different.  Subtract the

             equations if the signs of the coefficients of the variables are the

             same.

Step 4:  Solve the resultant equation for the remaining variable.

Step 5:  Select one equation from the original two equations, substitute the

             value of the variable found in step 4, and solve for the other

             variable.

A half  plane is the set of points on the Cartesian plane that are on one side of a line.

Procedure for Using Linear Programming

Step 1:  Write the objective function.

Step 2:  Write the constraints.

Step 3:  Graph the constraints.

Step 4:  Find the vertices of the polygonal region.

Step 5:  Substitute the coordinates of the vertices into the objective function

             and find the maximum or minimum value.

(Note:  The solutions will not always be integers.)

A relation is a set of ordered pairs of elements.

A relation is a function if for each element in the domain there is a unique element in the range.

The Vertical Line Test for Functions

If no vertical line can be intersect the graph of a relation of more than one point, then the relation is a function.

Procedure  for Graphing a Linear Function

Step 1:  Select at least 3 values for x.  (Only two values are necessary.  The

             third is used as a check.)

Step 2:  Substitute them in the function and find the corresponding values for

             f(x).

Step 3:  Plot the points (x, f(x)) on the Cartesian plane using the y axis as the

             f(x) axis.

Step 4:  Draw a line through the points.

An equation of the form f(x) = ax(2) + bx + c, where a, b, and c are  real numbers and a (= with a slash through it) 0 is called a quadratic function.  This equation can also be written as y = ax(2) + bx + c.  The graph of a quadratic function is called a parabola.

Procedure for Graphing the Quadratic Function y = ax(2) + bx + c

Step 1:  Identify a, b, and c, then find the vertex using x = -b/2a to get the x

            coordinate.  Then substitute this value in the equation y = ax(2) + bx +

            c to get the y coordinate.

Step 2:  Find the y intercept by substituting x = 0 in the equation and solving

             for y or use (0, c).

Step 3:  Find the x  intercepts by substituting 0 for y and solving the equation

             for x, either by factoring or by using the quadratic formula.

Step 4:  Find several other points in order to determine the shape.

Step 5:  Determine whether the parabola opens upward (a > 0) or downward

             (a < 0).  Plot the points and draw a smooth curve through the points.

An exponential function has the form f(x) = a(x), where a and x are real numbers such that a > 0 but a (= with a slash through it) 1.

Percent means hundredths or part of a hundred; i.e., 1% = 1/100.

Converting Percents to Decimals

In order to change a percent to a decimal, drop the % sign and move the decimal point two place to the left.

Converting Percent to Fractions

A percent can be converted to a fraction by dropping the percent sign and using the percent number as the number of a fraction whose denominator is 100.

Converting a Decimal to a Percent

To change a decimal to a percent, move the decimal point two places to the right and add a percent sign.

Changing a Fraction to a Percent

To change a fraction to a percent, first change the fraction to a decimal, and then change the decimal to a percent.

The three types of percent problems can be solved by this formula:

Part = Rate x Base (P = R x B).

I.  Finding a Part

To find a part, change the percent to a decimal or fraction and multiply.  Use P = R x B.

II.  Finding a Percent

To find what percent one number is of another number, substitute in the formula P = R x B and solve for R.  Be sure to change the decimal into a percent.

III.  Finding a Base

To find a base when a percent of it is known, substitute in the formula P = R x B and solve for B.  Be sure to change the percent to a decimal or fraction before dividing.

Interest is the fee charged or paid for the use of money.

The principal is the amount of money borrowed or placed in a savings account.

Rate is the percent of the principal that is paid for the use of the money.  (Rates are usually given for a year.)

Time or term is the duration that the money is borrowed or invested for or has been invested.  When the time is given in days or months, it must be converted to years by dividing by 365 or 12, respectively.

Maturity value is the amount of the loan or investment or savings (principal) plus the interest.

Formulas for Computing Simple Interest and Maturity Value

Interest = Principal x Rate x Time or I = PRT

Maturity Value = Principal + Interest or MV = P + I or MV = P(1+RT)

Formula for Computing Compound Interest

MV = P(1+r/n)(nt), where MVis the maturity value (Principal + Interest)

r = the yearly interest rate

n = number of periods the interest is compounded per year

t = term of the loan in years.

The effective rate (also known as the annual yield) is the simple interest rate which would yield the same maturity value over one year as the compound interest rate.

Formula for Effective Interest Rate

E = (1+r/n) (n) - 1, where

E = effective rate

n = number of periods per year the interest is calculated

r = interest rate per year (i. e., stated rate).

Formula for Finding the Future Value of an Annuity

FV = P(1+R(N) - 1/R), where

FV is the future value of the annuity

P is the payment

R is the interest rate per period (year, quarter, etc.)

N is the number of payments (periods in a year times the number of years).

Procedure for the Average Daily Balance Method

Step 1:  Find the balance for each transaction.

Step 2:  Find the number of days for each balance.

Step 3:  Multiply the balances by the number of days and find the sum.

Step 4:  Divide the sum by the number of days in the month.

Step 5:  Find the finance charge (multiply the average daily balance by the

             monthly rate).

Step 6:  Find the new balance (add the finance charge to the balance as of the

             last transaction).

The Constant Ratio Formula for APR

APR = 2Nt/P(T+1), where

N = number of payments per year (usually 12 since most loans are paid back in monthly payments)

I = finance charge (i.e., total interest plus any additional charges)

P = principal

T = total number of payments

Formula for the Rule of 78s

u = fk(k+1)/n(n+1), where

u = unearned interest (i.e., amount saved)

f = finance charge

k = number of remaining monthly payments

n = original number of payments

Procedure for Finding the Monthly Payment for a Fixed-Rate Mortgage

Step 1:   Find the down payment.

Step 2:  Subtract the down payment from the cost of the home to find the

             principal of the mortgage.

Step 3:  Divide the principal by 1000.

Step 4:  Find the number in the table that corresponds to the corresponds to

             the interest rate and the term of the mortgage.

Step 5:  Multiply that number by the number obtained in step 3 to get the

             monthly payment.

Procedure for Computing an Amortization Schedule

Step 1:   Find the interest for the first month.  Use I = PRT, where T = 1/12.

             Enter this value in the column labeled interest.

Step 2:  Subtract the interest from the monthly payment to get the amount

             paid on the principal.  Enter this amount in the column labeled

             Payment on Principal.

Step 3:  Subtract the amount of the payment on principal found in step 2 from

             the principal to get the balance of the loan.  Enter this in the column

             labeled Balance of Loan.

Step 4:  Repeat the steps using the amount of the balance found in step 3 for

             the new principal.

The markup (M) for an item is the difference between the cost and the selling price of an item.

The cost (C) of an item is the price that the merchant pays for the item.

The selling price (S) of an item is the price for which the merchant sells the item.

The basic formulas are:

Markup = Selling price - Cost or M = S - C

Selling Price = Cost + Markup or S = C + M

Cost = Selling price - Markup or C = S - M

Markup rate on selling price:  markup rate on cost/1 + markup rate on cost

Markup rate on cost:  markup rate on selling price/1 - markup rate on selling cost

Be sure to convert the percents to decimals before substituting in the formulas.

Two rays with a common endpoint form an angle.  The rays are called the sides of the angle and the endpoint is called the vertex.

An acute angle has a measure between 0 degrees and 90 degrees.

A right angle has a measure of 90 degrees.

An obtuse angle has a measure between 90 degrees and 180 degrees.

A straight angle has a measure of 180 degrees.

Two angles are called adjacent angles if they have a common vertex and a common side.

Two angles are said to be complementary if the sum of their measures is 90 degrees. 

Two angles are said to be supplementary if the sum of the measures of each is equal to 180 degrees.

The opposite angles formed by two intersecting lines are called vertical angles.  The measures of vertical angles are equal.

Alternate interior angles are the angles formed between two parallel lines on the opposite sides of the transversal that intercepts the two lines.  Alternate interior angles have equal measures.

Alternate exterior angles are the opposite exterior angles formed by the transversal that intersects two parallel lines.  Alternate exterior angles have equal measures.

Corresponding angles consist of one exterior and one interior angle on the same side of the transversal that intersects two parallel lines.  Corresponding angles have equal measures.

A triangle is a closed geometric figure consisting of three sides and three angles.

An isosceles triangle has two sides of equal length.

An equilateral triangle has three sides of equal length.

A scalene triangle has no two sides of equal length.

An acute triangle has three acute angles.

An obtuse triangle has one obtuse angle.

A right angle has one right angle.

The sum of the measures of the angles of a triangle is 180 degrees.

The Pythagorean theorem states that for any right triangle, the sum of the squares of the length of the legs of a right triangle is equal to the square of the length of the hypotenuse (the side opposite the right angle).

If triangle ABC is similar to triangle A'B'C', then

length of side AB/length of side A'B' = length of side AC/length of side A'C' = length of side BC/length of side B'C'

The sum of the measures of the angles of a polygon with n sides is (n - 2)180 degrees.

A trapezoid is a quadrilateral that has only two parallel sides.

A parallelogram is a quadrilateral in which opposite sides are parallel and equal in measure.

A rectangle is a parallelogram with four right angles.

A rhombus is a parallelogram in which all sides are equal in length.

A square is a rhombus with four right angles.

A circle is the set of all points in a plane equidistant from a fixed point called the center.

The formula for the circumference of a circle is C = 2(Pi)r or C = (Pi)d.

The formula for the area of a circle is A = (Pi)r(2).

sin A = length of side opposite angle A/length of hypotenuse = a/c

cos A = length of side adjacent to angle A/length of hypotenuse = b/c

tan A - length of side opposite angle A/length of side adjacent to angle A = a/b

Three steps can be used to solve right triangle trigonometric problems:

Step 1:   Draw and label the angles of the right triangle and the measures of

              the sides.

Step 2:   Select the appropriate formula and substitute the values (you may

              have to use the table in Appendix B).

Step 3:   Solve the equation for the unknown.

The angle of elevation of an objectr is the measure of the angle from a horizontal line at the point of an observer along the ine of sight to the object.  The angle of depression is the measure of an angle from a horizontal line at the point of an observer downward along the line of sight to the object.

The network is traversible if it is possible to pass through or trace each path exactly once without your pencil.  A vertex can be crossed more than once.

A probability experiment is a process that lead to well-defined results called outcomes.  An outcome is the result of a single trial of a probability experiment.

A sample space is the set of all possible outcomes of a probability experiment.

An event is any subset of the sample space of a probability experiment.

Formula for Classical Probability

The probability of any event E is

number of outcomes in E/total number of outcomes in the sample space S,

this probability is denoted by

P(E) = n(E)/N(S)

This probability, called classical probability, is based on a sample space S.

For any event, E, P(E) = 1 - P(E) (line over E).

A tree diagram consists of branches corresponding to the outcomes of two or more probability experiments that are done in sequence.

The formulas for odds are

odds in favor = P(E)/1 - P(E)

odds against = P(E)/1 - P(E) (line over E)

where P(E) is the probability that event E occurs and P(E) (line over E) is the probability that the event E does not occur.

If the odds in favor of an event E are a:b, then the probability that the event will occur is

P(E) = a/a+b

If the odds against an event E are c:d, then the probability that E will not occur is

P(E) (line over E) = c/c+d

The expected value for the outcomes of a probability experiment is

E = x(1) x P(x(1)) + x(2) x P(x(2)) + . . . + x(n) x P(x(n))

where the x's correspond to the outcomes and the P(x)'s are the corresponding probabilities of the outcomes.

Two events are  mutually exclusive if they cannot occur at the same time (i.e., they have no outcomes in common).

Addition Rule 1

When two events are mutually exclusive, the probability that A or B will occur is

P(A or B) = P(A) + P(B)

Addition Rule 2

If A and B are not mutually exclusive, then

P(A or B) = P(A) + P(B) - P(A and B)

Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring.

When the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability of the second event is changed, the events are said to be dependent.

Multiplication Rule 1

When two events are independent, the probability of both occurring is

P(A and B) = P(A) x P(B)

Multiplication Rule 2

When two events are dependent, the probability of both occurring is

P(A and B) = P(A) x P(B)

Formula for Conditional Probability

The probability that the second event B occurs given that the first event A has occurred can be found by dividing the probability that both events occurred by the probability that the first event has occurred.  The formula is

P(B/ A) = P(A and B)/P(A)

Fundamental Counting Rule

In a sequence of n events in which the first event can occur in k(1) ways and the second event can occur in k(2) ways and the third event can occur in k(3) ways and so on, the total number of ways the sequence can occur is

k(1), k(2), k(3), . . ., k(n)

For any natural number n

n! = n(n - 1)(n -2)(n - 3). . .3 x 2 x 1

n! is read as "n factorial."

0! is defined as 1.  (Note that this will be explained later.)

An arrangement of n distinct objects in a specific order is called a permutation of the objects.

Permutation Rule

The arrangement of n objects in a specific order using r objects at at time is called a permutation of n objects taking r objects at a time.  It is written as nPr and the formula is

nPr = n!/(n-r)!

A selection of objects without regard to order is called a combination.

Combination Rule

The number of combinations of r objects selected from n objects is denoted by nCr and is given by the formula

nCr = n!/(n-r)!r!

Data are measurements or observations that are gathered for an event under study.

Statistics is the branch of mathematics that involves collecting, organizing, summarizing, and presenting data and drawing general conclusions from data.

A population consists of all subjects under study.

A sample is a representative subgroup or subset of the population.

The mean is the sum of the values divided by the total number of values.  The symbol X (with a line over it) represents the mean.

X (with a line over it) = x(1) + x(2) + x(3) + . . . + x(n)/n = M(sideways) X/n

Formula for finding the mean for grouped data:

X (with a line over it) = M (sideways) f x X(m)/ n

where

f= frequencey

X(m) = midpoint

n = M (sideways)f or sum of the frequencies

The median is the midpoint of the data array.  The symbol for the median is MD.

The value that occurs most often in a data set is called the mode.

The range is the highest value minus the lowest value.  The symbol R is used for the range,

R = highest value - lowest value

Procedure for Finding the Variance and Standard Deviation

Step 1:   Find the mean.

Step 2:  Subtract  the mean from each data value in the data set.

Step 3:  Square the differences.

Step 4:  Find the sum of the squares.

Step 5:  Divide the sum by n - 1 to get the variance, where n is the number

             data values.

Step 6:  Take the square root of the variance to get the standard deviation.

The variance is an approximate average of the squares of the distance each value is from the mean.  The symbol for the variance is s(2).  The formula  for the variance is

s(2) = M(sideways)(X - X(with a line over it)(2)/n - 1

where

X = individual value

X(with a line over it) = mean

R = sample size

The standared deviation is the square root of the variance.  The symbol for the standard deviation is s.  The corresponding formula for the standard deviation is

s = (the square) s(2) = (the square) M(sideways)(X - X(with a line over it)(y(2)/n - 1

A percentile or percentile rank, P, of a data value indicates the percent of data values that are below the given data value.

A normal distribution is a continuous, symmetric, bell-shaped distribution.

Summary of the Properties of a Normal Distribution

1.  It is bell-shaped.

2.  The mean, median, and mode are equal and located at the center of the

     distribution.

3.  It is unimodal (i.e., it has only one mode).

4.  It is symmetrical about the mean, which is equivalent to saying that its

     shape is the same on both sides of a vertical line passing through the

     center.

5.  It is continuous--i.e., there are no gaps or holes.

6.  It never touches the x axis.  Though the curve gets increasingly close to

     the x axis, the two will never touch.

7.  The total area under a normal distribution curve is equal to 1.00, or 100%.

     This feature may seem unusual, since the curve never touches the x axis,

     but this fact can be proven mathematically by using calculus. 

8.  The area under a normal curve that lies within one standard deviation of

     the mean is approximately 0.68, or 68%; within two standard deviations,

     about 0.95 or 95%; and within three standard deviations, about 0.997, or

     99.7%.

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

A scatter plot is a graph of the ordered pairs (x, y) of the data values for two variables.

The correlation coefficient is a value that is computed from the paired data values in order to determine the strength of a relationship between two variables.  The symbol for the correlation coefficient computed from data obtained from a sample is r.

Formula for Finding the Value of r:

r = n(M(sideways) xy) - (M(sideways)x)(M(sideways)y)/(the square)[n(M(sideways)x(2)) - (M(sideways)x)(2)][n(M(sideways)y(2)) - (M(sideways)y)(2)]

where

n = the number of data pairs

M(sideways)x = the sum of the x values

M(sideways)y = the sum of the y values

M(sideways)xy = the sum of the products of the x and y values for each pair

M(sideways)x(2) = the sum of the squares of the x values

M(sideways)y(2) = the sum of the squares of the y values

Formulas for Finding the Values of a and b for the Equation of the Regression Line

b = n(M(sideways)xy) - (M(sideways)x)(M(sideways)y)/n(M(sideways)x(2)) - (M(sideways)x)(2) . . . slope

a = M(sideways)y - b(M(sideways)x)/n . . . y intercept