Study your flashcards anywhere!
Download the official Cram app for free >
 Shuffle
Toggle OnToggle Off
 Alphabetize
Toggle OnToggle Off
 Front First
Toggle OnToggle Off
 Both Sides
Toggle OnToggle Off
 Read
Toggle OnToggle Off
How to study your flashcards.
Right/Left arrow keys: Navigate between flashcards.right arrow keyleft arrow key
Up/Down arrow keys: Flip the card between the front and back.down keyup key
H key: Show hint (3rd side).h key
A key: Read text to speech.a key
71 Cards in this Set
 Front
 Back
What is a factorial?

For a positive integer n we define n factorial as n∙(n1)∙(n2)∙...∙3∙2∙1
denoted n! 

What are the two basic principles of counting?

The Rule of Sum
The Rule of Product 

What is the Rule of Sum?

If a first task can be performed in m ways and a second task can be performed in n ways and the two tasks can not be performed at the same time, then performing either task can be accomplished in any way of m+n ways.


What does it mean when two tasks can not be performed at the same time?

They are mutually exlusive.


What if the tasks are not mutually exlusive?

When we have two tasks A and B, the number of ways to perform either task is n(A)+n(B)n(A⋂B)


What is the Rule of Product?

If a procedure can be broken down into first and second stages and if there are m possible outcomes for the first stage and from each of those outcomes there are n possible outcomes for the second stage, then the total procedure can be carried out, in the designated order, in mn ways.


What is an rpermutation?

An ordered sequence of r objects from the set X.


How do we define P(n,r)

The number of rpermutations chosen from n objects.


How do we calculate P(n,r)

P(n,r)=n∙(n1)∙(n2)∙...∙(nr+1)
OR n!/(nr)! 

What is an rcombination?

An unordered selection of r objects chosen from the set X.


What is C(n,r)?

The number of rcombinations of n elements.


How do we find C(n,r)?

C(n,r)=n!/[r!∙(nr)!]


.

.


How do we express probability?

A ratio describing the number of favourable occurences to the number of total possible occurences for some specified result of an observable event.


What is empirical probability?

Probability that is determined on the basis of conducting an experiment or examining data


What is the sample space?

The set of all possible outcomes


How do we find the empirical probability of an experiment?

An experiment with a number of possible outcomes is peformed n times. If event E occurs r times, the empirical probability, denoted Pe(E), says that E will occur on any given trial of the experiment is Pe(E)=r/n


What is the Law of Large Numbers?

If an experiment is performed repeatedly the empirical probability of a particular outcome more and more closely approximates a fixed number as the number of trials increases.


When are outcomes equally likely?

If each outcome of an experiment is as likely to occur as any other outcome


What is the probability of an event E, if all outcomes are equally likely?

P(E) = number of outcomes in E / number of outcomes in sample space
P(E)= n(E)/n(S) 

What values must P(E) fall between?

0≤P(E)≤1


What does it mean if P(E)=0?

Event E is impossible


What does it mean if P(E)=1?

Event E will always happen


What is the probability of E compliment?

E compliment (denoted E bar) is 1P(E)


If A and B are mutually exclusive events, how do we find P(A or B)?

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


If A and B are not mutually exclusive events, how do we find P(A or B)?

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


If A and B are independant events, how do we find P(A and B)?

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


What does P(AB) mean?

Find the probability of event A given that event B has already occured.


How do we find P(AB)?

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


What is odds in favour of an event?

The ratio of the number of favorable outcomes to unfavorable outcomes.


How can we find odds in favor of event E?

n(E):n(not E)


How can we find the odds in favor of E using probability?

P(E):P(not E)


If the odds in favor of E are a:b what is P(E)?

P(E)= a/(a+b)


What is expected value?

The longrun average value over repeated plays of a payoff from a game or event


How do we find the expected value?

If the outcomes of an experiments have values V1, V2, V3...Vn then the expected value is P1V1 + P2V2 +...+PnVn


What is a point?

A location in space that does not have length, width, or height.
We represent points with a dot, and label them with capital letters. 

What is a line?

A set of points in a straight unlimited length with no thickness or endpoints.
Any two points determine one and only one line. We denote the line formed by points A & B as AB with a line (<>) above it 

What are collinear points?

Two or more points on the same line are collinear.


What is the plane?

A set of points in a flat surface that has no thickness and no edges


What is a line segment?

Two points on a line and all the points between them; it has a definite start and end point.
We denote the line segment between points A & B as AB with __ above it 

What does it mean to bisect a line segment?

To divide the line into 2 parts of equal length.


What is the midpoint?

The point that bisects the line.


What are congruent lines?

Lines that have the same length.
Denoted with ≅ 

What are parallel lines?

Two lines in the plane are parallel if they have no points in common.
Denoted l  m If two lines are not parallel, they MUST intersect at some point 

Where do lines intersect?

At the point of intersection


What are concurrent lines?

Three lines that intersect at the same point


What is a ray?

The ray AB is point A and all the points on line AB that are on the same side of A as B is
Denoted AB with > overtop 

How is an angle formed?

When two rays or line segments have a common end point.
Denoted with ∠ 

What is the vertex?

The common endpoint of the angle.
In ∠BAC A is the vertex 

How do we find the measure of the angle?

We find the amount of rotation. Denoted m(∠BAC), measured in degrees
0<m(∠BAC)<360 

How many angles are formed by two rays?

Two  the interior angle and the exterior angle


What does ∠BAC usually refer to?

The smaller of the two angles (the interior) so m(∠BAC)≤180


What is a zero angle?

When the angle measures 0


What is an acute angle?

When the angle measures less than 90


What is a right angle>?

When the angle measures 90


What is an obtuse angle?

When the angle measures between 90 and 180


What is a straight angle?

When the angle measures 180


What is a reflex angle?

When the angle measures between 180 and 360


What are perpendicular lines?

Two lines that intersect to form 4 90 degree angles.
Denoted l⊥m 

What are congruent angles?

Two angles that have the same measure.


What are complementary angles?

Angles whose measures add up to 90


What are supplementary angles?

Angles whose measures add up to 180


What are adjacent angles?

Two angles that have hte same vertex and a common side, but no common interior points.


What are linear angles?

Angles that are adjacent and have two noncommon sides on the same line


What are vertical angles?

Angles formed by two intersecting lines, that are not a linear pair of angles


What is a transversal?

A line that intersects two other lines


What are corresponding angles?

Two nonadjacent angles on the same side as the transversal, one interior and one exterior


What can we say about the corresponding angles if the two lines cut by a transversal are parallel?

They are congruent


What are alternate interior angles?

Two nonadjacent interior angles on opposite sides of the transverasl, forming a Z
Alternate interior angles are congruent 

What does the Alternate Interior Angle Theorem say?

Two lines cut by a transversal are parallel if and only if a pair of alternate interior angles are congruent.


What is the angle sum of triangles?

The sum of the measures of the interior angles of a triangle is 180
