Understanding Probability & Geometric Figures

Probability is a branch of mathematics that emerged in Italy and France in the sixteenth and seventeenth centuries. The theory of mathematical probability was founded by Blaise Pascal and Pierre Fermat.*Probability can be defined as the measure of how likely a certain outcome is for a certain event. Life insurance companies use probability to estimate how long a person is likely to live, doctors use probability to predict the success of treatment , and meteorologists use probability to forecast weather conditions. Collecting, organizing, describing, displaying, and making predictions on the basis of this data are skills that are increasingly important in society based on technology and communication.*The basic terms used for probability are:Experiment: A situation involving chance or probability that leads to results.Outcome: A result of an experiment.Sample Space: The set of all outcomes of an experiment.*E.T. Bell, Men of Mathematics (New York: Simon and Schuster, 1965).*National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics (Reston, VA: NCTM, 1989).

If all the outcomes of a sample space S are equally likely, the probability of even E can be written as:P(E) = number of outcomes in E number of outcomes in SExample:Experiment: Rolling a six-sided dieSample Space: 1, 2, 3, 4, 5, 6Event E: Rolling an even (2, 4, 6)So, P(E) = 3/6 or 1/2

If an event has a probability of 0, then the event is impossible.Example:Rolling a six-sided die and obtaining a number greater than 6.*Probability is never less than 0 or greater than 1.

If an event has a probability of 1, then the event is certain.Example:Rolling a six-sided die and obtaining a number less than 7.*Probability is never less than 0 or greater than 1.

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): P(A u B) = P(A) + P(B) - P(A n B)*Notice that if A and B are mutually exclusive, P(A n B) = 0

When two events share nothing in common, these events are said to be mutually exclusive.Example:Rolling a die and tossing a coin.NOT an example:Rolling an even number and rolling a 4.

If event A is complementary to event B, then P(A) + P(B) = 1. Example:Drawing a red card and drawing a black card.NOT an example:Drawing a face card and drawing an Ace.

Odds can be defined as ratios (part to part).The odds in favor of event E is the ratio of 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 6 in a single roll of a die.- There are only 2 possible ways to get a 5 or a 6 and 4 ways to NOT get a 5 or a 6. So, the odds of rolling a 5 or 6 are 2:4 or, when reduced, 1:2.The odds against event E are the opposite of the odds in favor.Example:The odds against rolling a 5 or a 6 in a single roll of a die are 4:2 or, when reduced, 2:1.

A single stage experiment is an experiment where only one event is being tested.Examples:Rolling a die, flipping a coin, picking a card, etc.

Theoretical probability refers to an "ideal" condition/situation in which an experiment is not performed to determine the likelihood of an outcome.*In fields like insurance, theoretical probability is the only method that can be used.

Experimental probability refers to the act of performing an experiment to determine the likelihood of an outcome.*A simulation is used to determine experimental probabilities.The basis for using simulations to approximate probabilities is the Law of Large Numbers.The law states that the more times a simulation is carried out, the closer the experimental probability is to theoretical probability.

A multistage experiment is an experiment where there is more than one action in the experimental process.Example:Roll a die, then flip a coin.

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.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 =-1.975So, this would not be a good risk because the expected value is less than what you originally spent. In other words, the reward is lower than the cost.

A permutation is a convenient way of determining the number outcomes for a situation with a large amount of outcomes. The method used to perform a permutation deals with something called a factorial.n factorial (n!) = n x (n - 1)...n x 1Example:5! = 5 x 4 x 3 x 2 x 1To figure out a permutation, one must figure out how many ways one can select "r" items or people from "n" items or people. The formula is written as: P(n,r) = n! / (n - r)!Example:P(5,2) = 5! / (5 - 2)! 5! / 3! 5 x 4 = 20*Permutations are only used when order matters.*For permutations, 0! = 1.

A combination is a convenient way of determining the number of outcomes for a situation in which order does NOT matter.To figure out a combination, one must figure out how many ways one can select "r" items or people from "n" items or people. The formula is written as: C(n,r) = n! / r! (n - r)!Example:C(5,2) = 5! / 2! (5 - 2)!5! / 2!3!20 / 2 = 10*Note: 2!3! is NOT the same as 2 x 6.

If an event A can occur in M ways and then event B can occure in N ways, then event A followed by event B can occure in M x N ways.Example:Roll a die, then flip a coin. How many ways are there to obtain an even number and a heads or tails?3 ways to get an even number (2, 4, 6)2 ways to get a heads or tails (1 head or 1 tail)3 x 2 = 6So, there are 6 ways to obtain an even number and either a heads or tails when rolling a die, then flipping a coin.

When determining P(event A, then event B), the probability is equal to P(A) x P(B).Example:What is the probability of rolling a die and obtaining a 2 on the first roll, then rolling the die a second time and obtaining an even number?First time: 1 in 6 chance of rolling a 2.Second time: 3 in 6 chance of rolling an even number.1/6 x 3/6 = 3/36 = 1/12So, the probability of rolling a die and obtaining a 2 on the first roll, then rolling the die again and obtaining an even number is 1/12.

Geometry is one of the earliest branches of mathematics and it is said that geometry was used more than 5,000 years ago by the Egyptians and Babylonians in surveying and architecture. Geometric figures are the figures that are studied in geometry. Geometry is defined as the study of relationships among lines, angles, surfaces, and solids.**"Problems of the Month," Mathematics Teachers 80 (October, 1987).

Plane figures are figures that are found within the geometric plane.Including:TrianglesScalene triangle: no 2 sides have the same lengthIsosceles triangle: at least 2 sides have the same lengthEquilateral triangle: all three sides are congruentCirclesA 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 radius is a line segment from a point on the circle to its center. A chord is a line segment whose endpoints are both in the circle.A chord that passes through the center is a diameter. A line that intersects the circle in exactly one point is called a tangent.

Angles are formed by the union of two rays or by two line segments that have a common endpoint.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.

Points are dots or other symbols found on a line that break the line into segments.A line is 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 considered colinear.A plane is a set of points that is undefined. It desribes the area in which all figures, lines, and all other geometric shapes are categorized and illustrated in.

A polygon can be defined as a simple closed curve that is the union of line segments. Polygons are often classified according to their number of line segments and sides. The endpoints of the line segments are called the polygon's vertices. The polygonal region is the union of a polygon and its interior. Any line segment connecting one vertex of a polygon to a nonadjacent vertex is called a diagonal.Special Named PolygonsTrapezoid: one pair of opposite sides are parallelIsosceles Trapezoid: non-parallel sides are congruentRhombus: opposite sides are parallel and all sides are congruentParellelogram: pairs of opposite sides are parallel and congruent Rectangle: pairs of opposite sides are parallel and congruent; made up of all right anglesSquare: all sides are congruent and all are right anglesKite: quadrilateral with two distinct pairs of congruent adjacent sides; either convex or concave

An equilateral polygon has all congruent sides. An equiangular polygon has all congruent angles. A regular polygon is both equilateral and equilangular.Polygon Theorems:The sum of the exterior angles of a convex polygon is 360 degrees.The sum of the interior angles of a convex n-gon is (n - 2) x 180 degrees.The sum of the interior angles of any n-gon is (n - 2) x 180 degrees.In a regular n-gon:Each interior angle's measure = (n - 2) x 180 degrees / n.Each exterior angle's measure = 360 degrees / n.Each central angle's measure = 360 degrees / n.

A tessellation is any arrangement in which nonoverlapping figures are placed together to entirely cover a region.*Regular hexagons, squares, and equilateral triangles are the only regular polygons that will tessellate.

Space figures are the geometrical figures that are 3-dimensional.A dihedral angle is an angle created between two intersecting planes.A surface is considered to be convex if the line segmenet that joins any two points on the surface contains no point that is in the region exterior to the surface.

Joining plane polygonal regions from edge to edge forms a surface called a polyderon.A regular polyhedron must:Be convexHave congruent regular polygonal regionsHave the same number of faces meet at each vertex of the polyhedronPlatonic Solid is the name used to describe the only regular polyhedra in geometry. They are:CubeTetrahedronOctahedronIcosahedronDodecahedronEuler's Formula for Polyhedra:(Let V, F, and E denote the respective number of vertices, faces, and edges of a polyhedron.) V + F = E + 2

Pyramids have a base with an apex that is opposite it, and lateral faces. They can be right pyramids or oblique pyramids. They can also be named by the shape of their base.

Prisms have 2 bases and lateral faces. They can be right or oblique. Like pyramids, the shape of their bases determine the name.

Cones and cylinders are not polyhedra because they are not composed of polygonal regions.Cones are generalized pyramids because they have a base, lateral surface, and an apex.Cylinders are generalized prisms because they have 2 bases and a lateral surface.

A symmetry of a plane figure is any rigid motion of the plane that moves all the points of the figure back to points of the figure.Examples:Flowers, animals, people, etc.

Reflection symmetry requires that for each point on the left side of a figure there is a corresponding point on the right side such that both points are the same perpendicular distance from the plane of symmetry.

Rotational symmetry is the term used to define a figure that can be turned about its center so it coincides with itself.

Translation symmetry requires a shape to be moved, but maintain its same dimensions.