Math 202 Mindmap

Probability is a relatively new branch of mathematics. It emerged in Italy and France during the sixteenth and seventeenth centuries from studies of strategies for gambling games. Life insurance companies use probability to estimate how long a person is likely to live, doctors use probability to estimate how long a person is likely to live, doctors use probability to predict the success of a treatment, and meteorologists use probability to forecast weather conditions. Collecting, organizing, describing, displaying, and interpreting data as well as making decisions and predictions on the basis of that information, are skills that are increasingly important in a society based on technology and communication.

A single stage event is when only one event is being experimented. Ex: Flipping one coin or spinning one spinner

A Theoretical Probability is the ideal condition that an outcome would take place and an experiment is not performed.Ex: The theoretical probability of rolling a 2 on a die is 1/6.- The basis for using simulations to approximate probabilities is the Law of Large Numbers.-The Law of Large Numbers states that the more times a simulation is carried out, the closer to the experimental probability is to the theoretical probability.-In many fields, like insurance for example, theoretical probability is the only way to go.

An event is said to be mutually exclusive when two events A and B share nothing in common.Example: -Rolling a die and tossing a coin-Rolling an even number and rolling an odd numberNot an Example:-Drawing a face card and drawing a red card-Rolling an even number and rolling a 4-An event that can be described using the intersection (and) , union (or) or complement (not) of other events is called a compound event.-Addition Property: (for unions/ or)P(AUB) = P (A) + P (B) - P(AnB)*notice that if A and B are mutually exclusive P(AnB) = 0

An event is said to be complementary when event A is complementary to event B. In other words one event does not affect or change the other event. Then P(A) + P(B) = 1.Example:-Rolling an even number and rolling an odd number-Drawing a black card and drawing a red cardNot Examples:-Rolling an even number and rolling a 3-Drawing a face card and drawing an ace

An experimental probability is when an experiment is performed in order to determine the likelihood of an outcome.Ex: Physically flipping a coin and recording the results-In other cases where theoretical probability is not always the answer we use Experiments.-For example, you could flip a coin to determine the gender of children. Or use a spinner to determine the probability that in a group of 5 people chosen at random at least 2 will share the same birthday.

If all of the outcomes of a sample space S are equally likely, the probability of event E is:P(E) = number of outcomes in E number of outcomes in SExample:-Experiment: Roll a six-sided die -Sample Space: 1,2,3,4,5,6 -Event E: Rolling an even (2,4,6)-P(E)= 3/6 or 1/2

If an event has a probability of 1, we say the event is certain.Ex: The probability that we will roll a six-sided die and get a number less than 7 is certain.

If an event has a probability of 0, we say that event is impossible.Ex: The probability that we will roll a six-sided die and get a number less than 1 is impossible.

Odds are ratios.-The odds in favor of event E is the ratio to the number of ways E can/will occur to the number of ways E can/will not occur.Example:The odds of rolling a 5 or a 6 in a single roll of a die. There are only 2 possible ways to get a 5 or 6 and 4 ways not to get a 5 or 6. So the odds of rolling a 5 or a 6 is 2:4. *Notice P(5 or 6) = 2/(2+4) = 6-On the other hand, the odds against event E is the ratio to the number of ways E cannot/will not occur to the number of ways E can/will occur. Example:The odds against rolling a 5 or a 6 in a single roll of a die is 4:2.An example in real life is the game roulette.

Multi Stage Experiments are ones that involve more than one step. For example, things such as drawing a card from a deck or flipping a coin and a die. Multiplication Principle-If event A can occur in m ways and then event B can ooccur in n ways (being careful to take into consideration the effect event A has on event B), then event A followed by event B can occur in m x n ways.Ex: Roll a die and then flip a coin, how many ways are the to get an even number and a head or a tail? 3 ways to get an even 2 ways to get a head or a tailSo 3 x 2= 6 ways Ex: How many ways are there to draw a King on the first card and a King on the second card without replacement? 4 ways to get a King the first draw only 3 ways to get a King on the secondSo 4 x 3= 12 waysEx: How many different outfits can you make out of 3 pairs of shorts and 4 tank tops?3 x 4= 12 different outfits-P(A and then B) = P(A) x P(B) taking into consideration the effect event A may have an event B (independent or dependent events).Ex: What is the probability of rolling a die and getting a 2 on the first roll and then rolling it a second time and getting an even number? P( 2 and then even) = P(2) x P(E) = 1/6 x 1/2= 1/12*This would be an example of an independent event, because the first roll has absolutely no affect on the second roll. Ex: A box contains 3 red marbles and 2 green marbles. A marble is randomly taken from the bag and not replaced, then a second marble is taken from the bag. What is the probability that the first marble is red and the second marble is green? P( R then G) = P(R) x P(G) = 3/5 x 2/4 = 6/20 = 3/10*This would be an example of a dependent event, because the second marble taken from the bag depends on the first marble taken from the bag.

The expected value = (P1 * V1) + (P2*V2) + . . . + (Pn*Vn) Where P1, P2. . . , Pn are the probabilities of each outcome and V1, V2, . . ., Vn are the values associated with each of the respective outcomes.For example: A lottery ticket has a scratch off spot with either "win" or "lose". There are 1000 tickets sold and only 5 have "win" under the scratch off spot. The ticket costs $2 and if you win you get $5. What is the expected value of playing this game?Event Probability Value P*V win 5/1000 3 5/1000*3= 15/1000lose 995/1000 -2 995/1000*-2 = -1990/1000 Sum = 15/1000 + -1990/1000 = -1975/1000 =-1.975 *This would not be a good risk because the number is less than what you spent. The goal is for the expected value to be as far right on the number line as possible. Acheiving 0 would mean that it was a fair game.

Permutations are used when sample spaces have too many outcomes to conveniently list. A permutation of objects is an arrangement of these objects in a particular order. n factorial is written as n! and means: n x n -1 x. . . x2 x1*Special case: 0! = 1Permutation Theorem: the number of permutations of n objects taken r objects at a time, where r is greater than or equal to 0 and r is less than or equal to n. nPr = n! (n-r) !Ex: In a school soccer league with seven teams, in how many ways can the teams finish in the positions winner, runner-up, and third place?-In forming all the possible arrangements for the three finishing places, order must be considered. For example, having Team 4, Team 7 and Team 2 in the positions of winner, runner-up, and third place respectively, is different from having Team 7, Team 2 and Team 4 in these three finishing spots. Using the multiplication principle there are 7 possibilities for the winner, and then 6 possibilities for the runner up, and then 5 possibilities are left for the third place. So, there are 7 x 6 x 5 different ways the teams can finish in the positions of winner, runner-up and third place. 7 x 6 x 5= 210 possibilitiesUsing the permutation formulafor 7 teams taken 3 at a time.7P3 = 7! = 7! = 7x6x5x4x3x2x1 = 7x6x5 (7-3) ! 4! 4x3x2x1 =210 possibilities

In permutations the order of the elements is important. However, in forming collection order is sometimes not important and can be ignored. A collection of objects for which order is not important is called a combination.Combination Theorem: The number of combinations of n objects taken r objects at a time, where r is less than or equal to 0 and r is less than or equal to n. nCr = n! (n-r) ! r!For example: The school chess club has 10 members. In how many ways can 3 members of the club be chosen for the Rules Committee?-There is no requirement to consider the order of the people on the 3-person Rules Committee. So the number of different committees can be found with the formula for combinations.10C3 = 10! = 10x9x8x7x6x5x4x3x2x1 (10-3) ! 3! (7x6x5x4x3x2x1) x (3x2x1) = 10x9x8 = 120 ways 6

"The study of relationships among lines, angles, surfaces, and solids is a major part of geometry, on of the earliest branches of mathematics. The word geometry is from the Latin geometria, which means earth-measure."(Probems of the Month, Mathematics Teacher 80 (October 1987): 550)

Points: are abstract ideas, which we illustrate by dots, corners of boxes, and tips of pointed objects. These concrete illustrations have width and thickness, but points have no dimensions. Line: a set of points that we describe as being "straight" and extending indefinitely in both directions. Arrows indicate that the line continues, indefinitely in both directions. If two or more points are on the same line, they are colinear.Plane: a plane is another set of points that is undefined. We describe it as being "flat" like the top of a table, but extending indefinitely. A plane can be illustrated by a drawing that uses arrows to indicate that it extends and is not bounded.Half-Planes: a line in a plane partitions the plane into three disjoint sets: the points on the line and two half-planes. Line Segments: consists of two endpoints on a line and all the points between them. -To bisect a line segment means to divide it into two parts of equal lengthHalf-Lines: a point on a line partitions the line into three disjoint sets: the point and two half-lines.

-If two lines intersect to form right angles they are Perpendicular.-If two lines are in a plane and they do not intersect, they are Parallel.-If two lines m and n are intersected by a third line t, we call line t a Transversal.-Two very special angles are created on the alternate sides of the transversal and interior to lines l and m. These angles are called alternate interior angles. If the two lines l and m are parallel, the alternate interior angles have the same measure. This forms the property of alternate interior angles: if two lines are intersected by a transversal, the lines are parallel if and only if the alternate interior angles created by the transversal have the same measure.

Ray: consists of a point on a line and all the points in one of the half-lines determined by the point.Angles: are formed by the union of two rays or by two line segments that have a common endpoint-This endpoint is called the vertex, and the rays or line segments are called the sides of the angle.The size of an agle is measured by the rotation required to turn one side of the anle to the other by pivoting about the vertex. The measure of an angle is given in degrees.-A protractor is used to measure angles. To measure an angle, place the center of the protractor on the vertex of the angle and line up one side of the angle with the baseline of the protractor.-An angle measured exactly 90 degrees is called a right angle.-If it is less than 90 degrees and greater than 0 degrees, it is called an acute angle.-If it is greater than 90 degrees and less than 180 degrees, then it is called an obtuse angle.-If it has a measure of exactly 180 degrees it is called a straight angle.-If it measures more than 180 degrees and less than 360 degrees then it is called a reflex angle.-If the sum of two angles is 90 degrees, the angles are called complementary. If their sum is 180 degrees, they are called supplementary.-If two angles have the same vertex, share a common side, and do not overlap, they are caled adjacent angles.-Nonadjacent angles formed by two intersecting lines are called vertical angles.

We can draw a curve through a set of points by using a single continuous motion. -A curve is called simple if while tracing the curve, the pencil never touches a point more than once (the curve does not crossover itself).-A curve is called closed if the pencil is lifted at the same point at which it started tracing.-If a curve satisfies both of these descriptions, then it is called a simple closed curve.-Jordan Curve Theorem: Any simple, closed curve partitions the points in a plane into three distinct regions: the interior, the exterior and the curve. -Convex and Concave Curves and Figures: a figure is convex if, and only if, it contains segement PQ for each pair of points P and Q contained in the figure. A figure that is not convex is concave.A simple way to determine if a shape is concave or convex is to imagine stretching an elastic band across the shape. If the elastic touches all points on the boundary the set is convex, if not it is concave. -Circle: a special case of a simple closed curve whose interior is a convex set. Each point on a circle is the same distance from a fixed point called the center. A line segment from a point on the circle to its center is a radius. A line segment whose endpoints are both on the circle is a chord. A chord that passes through the center is a diameter. A line that intersects the circle in exactly one point is called a tangent.

Polygon: a simple closed curve that is the union of line segments. -The union of a polygon and its interior is called a polygonal region.-Polygons are classified according to their number of line segments called sides. -The endpoints of these segments are called vertices. -Two sides of a polygon are adjacent sides if they share a common vertex, and two vertices are adjacent vertices if they share a common side.-Any line segment connecting one vertex of a polygon to a nonadjacent vertex is a diagonal.Special Named Polygons-Trapezoid: exactly one pair of opposite sides parallel-Isosceles Trapezoid: non-parallel sides congruent-Rhombus: opposite sides parallel and all sides of equal length-Parellelogram: pairs of opposite sides parallel and of equal length -Rectangle: pairs of opposite sides parallel and of equal length, and all right angles-Square: all sides of equal length and all right angles-Kite: quadrilateral with two distinct pairs of congruent adjacent sides, can be either convex or concaveDifferent Types of Triangles-Acute Triangle: all 3 acute angles-Right Triangle: contains one right angle-Equilateral Triangle: all 3 sides of equal length-Scalene Triangle: all 3 sides of different lengths-Isosceles Triangle: at least 2 sides of equal length-Obtuse Triangle: 1 angle obtuse

Quiz yourself on Polygons in this game!

Angles in Polygons-In spite of this range of possible sizes, there is a relationship between the sum of all the angles in a polygon and its number of sides. -In any triangle, the sum of the three angle measures is 180 degrees. -The sume of the angles in a polygon of four or more sides can be found by subdividing the polygon into triangles so that the vertices of the triangles are the vertices of the polygon. -An infinite variety of quadrilaterals can be formed, some convex, and others concave. However, since each quadrilateral can be partitioned into two triangles such that the vertices of the triangles are also the vertices of the quadrilateral, the sume of the angles of a quadrilateral will always be 360 degrees.Vertex Angle: is formed by two adjacent sides of the polygonCentral Angle: is formed by connecting the center of the polygon to two adjacent vertices of the polygonExterior Angle: is formed by one side of the polygon and the extension of an adjacent sideCongruence-Two figures are congruent if one can be placed on the other so that they coincide. Another way to describe congruent plane figures is to say that they have the same size and shape. -Two line segments are congruent if they have the same length, and two angles are congruent if they have the same measure.Regular Polygons1. All angles are congruent2. All sides are congruent-A polygon with all of its sides congruent is called equilateral.-A polygon with all of its angles congruent is called equiangular.Theorem: Angle Measure in a Regular n-gon,-each interior angle has a measure (n-2) * 180/n-each exterior angle has measure 360/n-each central angle has measure 360/n

Any arrangement in which nonoverlapping figures are placed together to entirely cover a region is called a tessellation.-The hexagonal cells of a honeycomb can be placed side by side with no uncovered gaps between them. -Floors and ceilings are often tessellated or tiled with square-shaped material, because squares can be joined together without gaps or overlaps. -Equilateral triangles are also commonly used for tessellations.These three types of polygons- regular hexagons, squares, and equilateral triangles are the only regular polygons that will tessellate.

Here you can actually build your own tessellation!

The notions of space in geometry is an undefined term, just as the ideas of point, line, and plane are undefined. We intuitively think of space as three-dimensional and of a plane as only two-dimensional.

Planes and Lines in Space-A plane in space partitions the points in space into three disjoint sets: the plane itself and two half-spaces.-Two planes in space can either be parallel or intersect in a straight line in space.-When two planes intersect eachother, there is an angle created between them. This angle is called a dihedral angle.

Surfaces and Solids-A surface is convex if the line segment that joins any two points on its surface contains no point that is in the region exterior to the surface. Polyhedra (plural of polyhedron): Joining plane polygonal regions from edge to edge forms a surface called a polyhedron.-Each of the polygonal regions is called a face.-The faces intersect with vertices and edges of the polyhedron.-The union of a polyhedron and its interior is called a solid.-Generally, polyhedra are named according to the number of faces it has: tetrahedron has four faces, pentahedron has five, hexahedron has six. Some polyhedra are quite beautiful and complicated- they have fancier, longer names. -A net is a 2-dimensional drawing that can be cut out and folded to create a polyhedron.-There are only 5 regular polyhedra. As group, they are called the platonic solids. Cube 6 squaresTetrahedron 4 equilateral trianglesOctahedron 8 equilateral triangles Icosahedron 20 equilateral trianglesDodecahedron 12 regular pentagons Semiregular Polyhedra: polyhedra whose faces are two or more regular polygons with the same arrangement of polygons around each vertex.Pyramids: have a base with an apex that is oppisite it, and lateral faces. The base of a pyramid can be any polygon, but its sides are always triangular. Pyramids are named according to the shapes of their bases. Pyramids whoses sides are isosceles triangles are called right pyramids. Otherwise, the pyramid is called an oblique pyramid.Prism: has two parallel bases, upper and lower, which are congruent polygons. Like pyramids, prisms get their names from the shape of their bases. If the lateral sides of a prism are perpendicular to the bases they are rectangles. Such a prism is called a right prism. If some of the lateral faces of a prism are parallelograms that are not rectangles the prism is called an oblique prism.-The union of a prism and its interior is called a solid prism. A rectangular prism that is a solid is sometimes called a rectangular solid.-Therre is a formula that relates the numbers of vertices, edges, and faces of a polyhedron. This formula was first stated by Rene Descartes about 1635. In 1752 it was discovered again by Leonhard Euler and is now referred to as Euler's Formula.Euler's Formula for PolyhedraTheorem: Let V, F, and E denote the respective number of vertices, faces, and edges of a polyhedron.Then V + F = E + 2

Sphere: is the set of points in space that are the same distance from a fixed point, called the center. The union of a sphere and its interior is called a solid sphere. -A line segment joining the center of a sphere to a point on the sphere is called a radius.-The length of such a line segment is also called the radius of the sphere.- A line segment containing the center of the sphere and whose endpoints are on the sphere is called a diameter, and the length of such a line segment is called the diameter of the sphere.Cones and Cylinders-Cones and cylinders are the circular counterparts of pyramids and prisms. A cone has a circular region for a base and a lateral surface that slopes to the vertex (apex). -If the vertex lies directly above the center of the base, the cone is called a right cone or usually just a cone, otherwise its is an oblique cone. -Cylinder: has two parallel circular bases of the same size and a lateral surface that rises from one base to the other.

-If a line can be drawn through a figure so that each point on one side of the line has a matching point on the other side at the same perpendicular distance from the line, it is a line of symmetry.-If two points on opposite sides of this line match up, one is called the image of the other.-Figures that seem to have the form and balance of symmetric figures, but have no lines of reflection may have rotation symmetry. This means that it can be turned about its center so that it coincides with itself. The Center of Rotation is the fixed point around which a two-dimensional figure is rotated. A two-dimensional figure always rotates around a point and a three-dimensional figure rotates around an axis.Plane of Symmetry: the points on one side of a plane are the reflection of the points on the other side. Reflection Symmetry: for figures in space can be defined mathematically by requiring that for each point on the left side there is a corresponding point on the right side such that both points are the same perpendicular distance from the plane of symmetry. Vertical Symmetry: the plane of symmetry is perpendicular to the ground. Axis of Symmetry: passes through the center from the top and to the base