The document discusses the history and development of probability theory. It begins by describing how the foundations of probability were established in the 17th century by Blaise Pascal and Pierre de Fermat through their study of games of chance. It then provides definitions and explanations of key probability concepts like sample space, events, and the probability of an event occurring. The document goes on to discuss different models for representing probability problems geometrically, including linear models, area models, and coordinate geometry models. It also covers famous probability puzzles and paradoxes like the Monty Hall problem, Bertrand's paradox, and the Sleeping Beauty problem. Finally, it briefly mentions some applications of probability in fields like quantum mechanics, polymer sciences, and
This document contains a collection of math problems and warm-ups covering topics like geometry, trigonometry, algebra, and calculus. There are over 30 problems presented without solutions for students to work through. The problems involve finding areas and perimeters of shapes, solving equations, working with functions, and more.
The document provides an overview of the Pythagorean theorem presented by Tanya Van Dam Kalles. It discusses Pythagoras and the development of the theorem. Examples are given to demonstrate how to use the theorem to solve problems related to right triangles, including finding the distance from home plate to second base in baseball and calculating the height of a window using the length of a leaning ladder.
The document discusses the Pythagorean theorem and square roots. It begins by defining a right triangle and its components. It then states the Pythagorean theorem, which relates the lengths of the sides of a right triangle. An example problem demonstrates using the theorem to find the height of a wall. The document concludes by defining the square root, explaining that the square root of a number is its positive root and how to calculate and approximate square roots.
The document discusses basic geometrical shapes and formulas for calculating their perimeters. It defines a loop or polygon as a shape formed when the ends of a rope or string resting in a plane are connected. Plane shapes enclosed by straight lines are called polygons, including triangles, rectangles, and squares. Formulas are provided for calculating the perimeters of triangles, rectangles, squares, and combinations of shapes. Examples are included to demonstrate calculating total perimeter lengths required for fencing or roping off areas.
Mia Mia is a real time local search engine that enables people to search for a search provider anywhere with ease and convenience. Mia Mia is one of the best listing website for MBA Classes in Mumbai. We are also known for our systematic listing of various IPCC, Science coaching for CBSE, Engineering and other courses in Mumbai. QLI is a class where each student is our priority. Top MBA Institutes in Mumbai for CAT, XAT, NMAT and IIFT are listed on MiaMia.For details - visit: http://miamia.co.in/
The document discusses basic geometric shapes and formulas for calculating their perimeters. It defines a loop or polygon as a shape formed by connecting line segments end to end. Triangles have three sides and their perimeter is calculated as the sum of the three side lengths. Specific types of triangles like equilateral triangles are discussed. Rectangles are four-sided polygons with right angles, and squares are rectangles with four equal sides. Formulas are provided for calculating the perimeters of squares and rectangles based on their side lengths. Some example problems demonstrate applying these concepts and formulas to calculate perimeters of fenced or roped areas composed of multiple shapes.
The document provides 3 examples of perimeter calculations: 1) Finding the perimeter of a triangle with sides of 5, 9, and 11 cm, which equals 25 cm. 2) Calculating the perimeter of a rectangle and two similar squares shown in a diagram, which equals 34 cm. 3) Determining the cost of barbed wire to fence a 120m x 120m square garden, which is RM5280 using 480m of fencing at RM11 per meter.
This document discusses the Pythagorean theorem. It begins with an introduction of the theorem, followed by examples of using it to solve for the length of a triangle's hypotenuse given the lengths of the other two sides. The document then provides task cards with real world problems for students to solve using the Pythagorean theorem. It concludes with a post-test asking students to use the theorem to find hypotenuse lengths.
This document contains a collection of math problems and warm-ups covering topics like geometry, trigonometry, algebra, and calculus. There are over 30 problems presented without solutions for students to work through. The problems involve finding areas and perimeters of shapes, solving equations, working with functions, and more.
The document provides an overview of the Pythagorean theorem presented by Tanya Van Dam Kalles. It discusses Pythagoras and the development of the theorem. Examples are given to demonstrate how to use the theorem to solve problems related to right triangles, including finding the distance from home plate to second base in baseball and calculating the height of a window using the length of a leaning ladder.
The document discusses the Pythagorean theorem and square roots. It begins by defining a right triangle and its components. It then states the Pythagorean theorem, which relates the lengths of the sides of a right triangle. An example problem demonstrates using the theorem to find the height of a wall. The document concludes by defining the square root, explaining that the square root of a number is its positive root and how to calculate and approximate square roots.
The document discusses basic geometrical shapes and formulas for calculating their perimeters. It defines a loop or polygon as a shape formed when the ends of a rope or string resting in a plane are connected. Plane shapes enclosed by straight lines are called polygons, including triangles, rectangles, and squares. Formulas are provided for calculating the perimeters of triangles, rectangles, squares, and combinations of shapes. Examples are included to demonstrate calculating total perimeter lengths required for fencing or roping off areas.
Mia Mia is a real time local search engine that enables people to search for a search provider anywhere with ease and convenience. Mia Mia is one of the best listing website for MBA Classes in Mumbai. We are also known for our systematic listing of various IPCC, Science coaching for CBSE, Engineering and other courses in Mumbai. QLI is a class where each student is our priority. Top MBA Institutes in Mumbai for CAT, XAT, NMAT and IIFT are listed on MiaMia.For details - visit: http://miamia.co.in/
The document discusses basic geometric shapes and formulas for calculating their perimeters. It defines a loop or polygon as a shape formed by connecting line segments end to end. Triangles have three sides and their perimeter is calculated as the sum of the three side lengths. Specific types of triangles like equilateral triangles are discussed. Rectangles are four-sided polygons with right angles, and squares are rectangles with four equal sides. Formulas are provided for calculating the perimeters of squares and rectangles based on their side lengths. Some example problems demonstrate applying these concepts and formulas to calculate perimeters of fenced or roped areas composed of multiple shapes.
The document provides 3 examples of perimeter calculations: 1) Finding the perimeter of a triangle with sides of 5, 9, and 11 cm, which equals 25 cm. 2) Calculating the perimeter of a rectangle and two similar squares shown in a diagram, which equals 34 cm. 3) Determining the cost of barbed wire to fence a 120m x 120m square garden, which is RM5280 using 480m of fencing at RM11 per meter.
This document discusses the Pythagorean theorem. It begins with an introduction of the theorem, followed by examples of using it to solve for the length of a triangle's hypotenuse given the lengths of the other two sides. The document then provides task cards with real world problems for students to solve using the Pythagorean theorem. It concludes with a post-test asking students to use the theorem to find hypotenuse lengths.
The document defines perimeter and provides formulas for calculating the perimeter of rectangles, squares, and triangles. It explains that perimeter is the distance around a two-dimensional shape. For rectangles, the perimeter formula is P=2(length+breadth). For squares, the formula is P=4xside. For triangles, the formula is P=a+b+c, where a, b, and c are the lengths of the three sides. An example problem calculates the perimeter of a triangular plot of land that is to be fenced with four rounds of wire.
The document discusses basic facts about triangles, including:
- A triangle is a three-sided polygon formed by three line segments.
- The three angles of any triangle always sum to 180 degrees.
- The triangle inequality states that the sum of any two side lengths must be greater than the third side length.
- Two triangles are congruent if they have the same shape and size.
The document defines perimeter as the total length of the sides surrounding a shape. It provides the formulas to calculate the perimeter of common shapes like triangles, squares, rectangles, pentagons, and circles. For each shape, it gives an example with measurements and calculates the perimeter. The perimeter of a triangle is the sum of the lengths of its three sides. The perimeter of a square is the sum of the lengths of its four equal sides. The perimeter of a rectangle is calculated by adding the lengths of all four of its sides.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It was named after the Greek mathematician Pythagoras, who lived in the 6th century BC. The theorem can be used to calculate unknown side lengths in right triangles. Some examples are also given to demonstrate applying the theorem.
Pythagoras was an ancient Greek thinker, but he was not the founder of the Pythagorean theorem. That honor goes to his followers, known as the Pythagorean Brotherhood, who established the theorem over 100 years after Pythagoras' death. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This relationship has many practical applications in fields like engineering, construction, physics, and astronomy that involve calculating distances.
This document provides a tutorial on using the Pythagorean theorem to solve for missing sides of right triangles. It explains the theorem and its formula, how to identify the sides of a right triangle, and how to solve for both missing hypotenuses and legs. Examples are worked through step-by-step and opportunities for practice are provided throughout the tutorial. The goal is for students to demonstrate mastery in solving for missing sides of right triangles.
The document explains the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. It provides examples of using the theorem to determine if a triangle is a right triangle, and to find the length of missing sides when the other two sides are known.
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object
The document discusses calculating the total surface area (TSA) of different 3D shapes by unfolding them into their 2D nets. It provides the TSA formulas and worked examples for rectangular prisms, triangular prisms, cylinders, and square pyramids. Engineers, builders and others use TSA to determine material needs like paint or carpet for projects. The document emphasizes that TSA is an area measurement, not a volume.
This document discusses Pythagoras' theorem and how to use it to solve problems involving right triangles. It explains that Pythagoras' theorem relates the lengths of the sides of a right triangle, where the hypotenuse is the side opposite the right angle. The learning objectives are to understand this relationship and how to use Pythagoras' theorem to find the length of an unknown side of a right triangle. Examples are provided for students to practice applying the theorem to solve problems.
Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The document explains how to use Pythagoras' theorem to find the length of any side of a right-angled triangle if the other two sides are known. It provides examples of calculating missing side lengths and notes that the theorem can be rearranged to find a non-hypotenuse side if the hypotenuse is known instead. Correct notation for showing working is emphasized.
Classify three-dimensional figures according to their properties.
Use nets and cross sections to analyze three-dimensional figures.
Extend midpoint and distance formulas to three dimensions
The document provides instructions on using trigonometric functions to find missing angles or sides of right triangles. It begins with reminders of trigonometric definitions and ratios. Examples are then given to demonstrate finding a missing side using the Pythagorean theorem or a trigonometric ratio, and finding a missing angle using an inverse trigonometric function. Tips are provided on determining which method to use based on the information given in a problem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The document explains the Pythagorean theorem and provides examples of using it to determine if a triangle is a right triangle or to find the length of a missing side.
The document discusses the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It provides examples of using the theorem to determine whether a triangle is right, acute, or obtuse based on the side lengths. It also discusses classifying triangles, finding missing side lengths, and explores Pythagorean triples.
This document contains explanations and examples regarding triangle theorems and properties including: the exterior angle theorem, isosceles triangle theorem, relationship between side length and angle size, triangle inequality theorem, and determining possible third side lengths given two sides of a triangle. Examples are provided to demonstrate applying these triangle concepts to determine largest angles, longest/shortest sides, and possible side lengths.
Pythagoras discovered that the ancient Egyptians used a 3:4:5 right triangle to build the pyramids. He investigated this further and deduced the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Pythagoras proved this by drawing squares on each side of right triangles and showing that the areas added up. The Pythagorean theorem has been used since ancient times in architecture, engineering, and more recently in technology like screens.
This document discusses formulas and strategies for computing the perimeter and area of basic shapes like triangles, parallelograms, trapezoids, and circles. It provides the formulas for finding the perimeter of squares, rectangles, and other polygons by adding the side lengths. Formulas are also given for finding the area of triangles, parallelograms, squares, rectangles, trapezoids, circles, and more complex irregular shapes by decomposing them into basic components. Examples are worked through on various pages and homework is assigned covering the material.
The document provides a lesson on the Pythagorean theorem. It includes examples of using the theorem to find the length of hypotenuses and legs of right triangles. It also gives practice problems and their step-by-step solutions. The document aims to teach students how to apply the Pythagorean theorem to solve problems involving right triangles.
The student is able to calculate geometric probabilities and use geometric probability to predict results in real-world situations. Geometric probability is based on a ratio of geometric measures such as length or area to calculate the probability of an event. Examples are provided to demonstrate calculating probabilities of events for points on lines and in plane figures, outcomes of experiments with spinners or stoplights, and areas of shapes within rectangles.
5.13.3 Geometric Probability and Changing Dimensionssmiller5
Students will
* Calculate geometric probabilities
* Use geometric probability to predict results in real-world situations
* Predict the effects of changing dimensions on the perimeter/circumference and area of a figure.
The document defines perimeter and provides formulas for calculating the perimeter of rectangles, squares, and triangles. It explains that perimeter is the distance around a two-dimensional shape. For rectangles, the perimeter formula is P=2(length+breadth). For squares, the formula is P=4xside. For triangles, the formula is P=a+b+c, where a, b, and c are the lengths of the three sides. An example problem calculates the perimeter of a triangular plot of land that is to be fenced with four rounds of wire.
The document discusses basic facts about triangles, including:
- A triangle is a three-sided polygon formed by three line segments.
- The three angles of any triangle always sum to 180 degrees.
- The triangle inequality states that the sum of any two side lengths must be greater than the third side length.
- Two triangles are congruent if they have the same shape and size.
The document defines perimeter as the total length of the sides surrounding a shape. It provides the formulas to calculate the perimeter of common shapes like triangles, squares, rectangles, pentagons, and circles. For each shape, it gives an example with measurements and calculates the perimeter. The perimeter of a triangle is the sum of the lengths of its three sides. The perimeter of a square is the sum of the lengths of its four equal sides. The perimeter of a rectangle is calculated by adding the lengths of all four of its sides.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It was named after the Greek mathematician Pythagoras, who lived in the 6th century BC. The theorem can be used to calculate unknown side lengths in right triangles. Some examples are also given to demonstrate applying the theorem.
Pythagoras was an ancient Greek thinker, but he was not the founder of the Pythagorean theorem. That honor goes to his followers, known as the Pythagorean Brotherhood, who established the theorem over 100 years after Pythagoras' death. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This relationship has many practical applications in fields like engineering, construction, physics, and astronomy that involve calculating distances.
This document provides a tutorial on using the Pythagorean theorem to solve for missing sides of right triangles. It explains the theorem and its formula, how to identify the sides of a right triangle, and how to solve for both missing hypotenuses and legs. Examples are worked through step-by-step and opportunities for practice are provided throughout the tutorial. The goal is for students to demonstrate mastery in solving for missing sides of right triangles.
The document explains the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. It provides examples of using the theorem to determine if a triangle is a right triangle, and to find the length of missing sides when the other two sides are known.
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object
The document discusses calculating the total surface area (TSA) of different 3D shapes by unfolding them into their 2D nets. It provides the TSA formulas and worked examples for rectangular prisms, triangular prisms, cylinders, and square pyramids. Engineers, builders and others use TSA to determine material needs like paint or carpet for projects. The document emphasizes that TSA is an area measurement, not a volume.
This document discusses Pythagoras' theorem and how to use it to solve problems involving right triangles. It explains that Pythagoras' theorem relates the lengths of the sides of a right triangle, where the hypotenuse is the side opposite the right angle. The learning objectives are to understand this relationship and how to use Pythagoras' theorem to find the length of an unknown side of a right triangle. Examples are provided for students to practice applying the theorem to solve problems.
Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The document explains how to use Pythagoras' theorem to find the length of any side of a right-angled triangle if the other two sides are known. It provides examples of calculating missing side lengths and notes that the theorem can be rearranged to find a non-hypotenuse side if the hypotenuse is known instead. Correct notation for showing working is emphasized.
Classify three-dimensional figures according to their properties.
Use nets and cross sections to analyze three-dimensional figures.
Extend midpoint and distance formulas to three dimensions
The document provides instructions on using trigonometric functions to find missing angles or sides of right triangles. It begins with reminders of trigonometric definitions and ratios. Examples are then given to demonstrate finding a missing side using the Pythagorean theorem or a trigonometric ratio, and finding a missing angle using an inverse trigonometric function. Tips are provided on determining which method to use based on the information given in a problem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The document explains the Pythagorean theorem and provides examples of using it to determine if a triangle is a right triangle or to find the length of a missing side.
The document discusses the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It provides examples of using the theorem to determine whether a triangle is right, acute, or obtuse based on the side lengths. It also discusses classifying triangles, finding missing side lengths, and explores Pythagorean triples.
This document contains explanations and examples regarding triangle theorems and properties including: the exterior angle theorem, isosceles triangle theorem, relationship between side length and angle size, triangle inequality theorem, and determining possible third side lengths given two sides of a triangle. Examples are provided to demonstrate applying these triangle concepts to determine largest angles, longest/shortest sides, and possible side lengths.
Pythagoras discovered that the ancient Egyptians used a 3:4:5 right triangle to build the pyramids. He investigated this further and deduced the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Pythagoras proved this by drawing squares on each side of right triangles and showing that the areas added up. The Pythagorean theorem has been used since ancient times in architecture, engineering, and more recently in technology like screens.
This document discusses formulas and strategies for computing the perimeter and area of basic shapes like triangles, parallelograms, trapezoids, and circles. It provides the formulas for finding the perimeter of squares, rectangles, and other polygons by adding the side lengths. Formulas are also given for finding the area of triangles, parallelograms, squares, rectangles, trapezoids, circles, and more complex irregular shapes by decomposing them into basic components. Examples are worked through on various pages and homework is assigned covering the material.
The document provides a lesson on the Pythagorean theorem. It includes examples of using the theorem to find the length of hypotenuses and legs of right triangles. It also gives practice problems and their step-by-step solutions. The document aims to teach students how to apply the Pythagorean theorem to solve problems involving right triangles.
The student is able to calculate geometric probabilities and use geometric probability to predict results in real-world situations. Geometric probability is based on a ratio of geometric measures such as length or area to calculate the probability of an event. Examples are provided to demonstrate calculating probabilities of events for points on lines and in plane figures, outcomes of experiments with spinners or stoplights, and areas of shapes within rectangles.
5.13.3 Geometric Probability and Changing Dimensionssmiller5
Students will
* Calculate geometric probabilities
* Use geometric probability to predict results in real-world situations
* Predict the effects of changing dimensions on the perimeter/circumference and area of a figure.
Areas related to circles - Areas of combinations of plane figures (Class 10 M...Let's Tute
Areas related to circles - Areas of combinations of plane figures (Class 10 Maths).
Let's tute is an E-school or E- platform which is free for the student.Students will watch "MATHS" Videos for conceptual understanding.
Contact Us -
Website - www.letstute.com
YouTube - www.youtube.com/letstute
1. The document defines geometric probability as probability based on ratios of geometric measures like length and area, where outcomes are represented by points or regions.
2. Examples are provided to demonstrate calculating geometric probabilities for situations like choosing a random point on a line segment or in a plane figure, the probability of light cycles, and spinners.
3. Additional examples find the probabilities of points chosen in a rectangle landing in specific shapes like a circle, trapezoid, or one of two squares.
This document appears to be a collection of blank pages with no discernible content. There is no information provided across the 11 pages to summarize. The document consists solely of page numbers without any text, images or other material that could be summarized.
The document describes a geometric distribution problem where a state trooper checks cars for speeding and the probability of catching a speeding teenager is 47%. The question asks for the probability that the first speeding teenager caught is the third car of the day. Using the geometric distribution formula of P(x)=(p)(q)^(x-1), where p is the probability of success (47%) and q is the probability of failure (1-p or 53%), the solution is calculated as P(3)=0.47*(0.53)^2=0.132023 or a 13.2% probability.
This document provides a lesson on geometric probability from a Holt Geometry textbook. It includes examples of calculating probabilities based on ratios of lengths, angles, and areas. It demonstrates geometric probability concepts for events involving points on line segments, locations on spinners, and areas within plane figures. The document concludes with a lesson quiz to assess understanding of calculating geometric probabilities.
This document contains 20 math word problems covering a variety of topics including: geometry (area of shapes, properties of circles, triangles, etc.), algebra (equations, functions, graphs), trigonometry, probability, and more. The problems range from basic calculations to more complex multi-step problems.
This report summarizes research on the motion of particles on curves. It was found that:
1) The center of mass of 3 points on an ellipse that divide its perimeter evenly traces out a smaller ellipse of the same shape.
2) The maximum product of distances between 4 particles on a rectangle occurs when particles are at the corners for small rectangles, but 2 particles move off the corners for larger rectangles.
3) The center of mass of n points on a square that divide its perimeter evenly traces out a smaller square n times for odd n, and remains fixed at the center for even n.
324 Chapter 5 Relationships Within TrianglesObjective To.docxgilbertkpeters11344
324 Chapter 5 Relationships Within Triangles
Objective To use inequalities involving angles and sides of triangles
In the Solve It, you explored triangles formed by various lengths of board. You may have
noticed that changing the angle formed by two sides of the sandbox changes the length
of the third side.
Essential Understanding Th e angles and sides of a triangle have special
relationships that involve inequalities.
Property Comparison Property of Inequality
If a 5 b 1 c and c . 0, then a . b.
For a neighborhood improvement project, you
volunteer to help build a new sandbox at
the town playground. You have two boards
that will make up two sides of the
triangular sandbox. One is 5 ft long and the
other is 8 ft long. Boards come in the
lengths shown. Which boards can you use
for the third side of the sandbox? Explain.
Inequalities in
One Triangle
5-6
t
t
tt
o
lele
f
Think about
whether the shape
of the triangle
would be easy to
play in.
Dynamic Activity
Triangle
Inequalities
T
A
C T I V I T I
E S T
AAAAAAAA
C
A
CC
I E
SSSSSSSS
DY
NAMIC
Proof of the Comparison Property of Inequality
Given: a 5 b 1 c, c . 0
Prove: a . b
Statements Reasons
1) c . 0 1) Given
2) b 1 c . b 1 0 2) Addition Property of Inequality
3) b 1 c . b 3) Identity Property of Addition
4) a 5 b 1 c 4) Given
5) a . b 5) Substitution
Proof
hsm11gmse_NA_0506.indd 324 3/6/09 11:56:15 AM
http://media.pearsoncmg.com/aw/aw_mml_shared_1/copyright.html
Problem 1
Got It?
Lesson 5-6 Inequalities in One Triangle 325
Th e Comparison Property of Inequality allows you to prove the following corollary to
the Triangle Exterior Angle Th eorem (Th eorem 3-11).
Proof of the Corollary
Given: /1 is an exterior angle of the triangle.
Prove: m/1 . m/2 and m/1 . m/3.
Proof: By the Triangle Exterior Angle Th eorem, m/1 5 m/2 1 m/3. Since
m/2 . 0 and m/3 . 0, you can apply the Comparison Property of
Inequality and conclude that m/1 . m/2 and m/1 . m/3.
Applying the Corollary
Use the fi gure at the right. Why is ml2 S ml3?
In nACD, CB > CD, so by the Isosceles Triangle Th eorem,
m/1 5 m/2. /1 is an exterior angle of nABD, so by the
Corollary to the Triangle Exterior Angle Th eorem, m/1 . m/3.
Th en m/2 . m/3 by substitution.
1. Why is m/5 . m/C?
You can use the corollary to Th eorem 3-11 to prove the following theorem.
Corollary Corollary to the Triangle Exterior Angle Theorem
Corollary
Th e measure of an exterior
angle of a triangle is greater
than the measure of each of
its remote interior angles.
If . . .
/1 is an exterior angle
Then . . .
m/1 . m/2 and
m/1 . m/3
2 1
3
Proof
3
4
1
25
A
CD
B
Theorem 5-10
Theorem
If two sides of a triangle are
not congruent, then the
larger angle lies opposite
the longer side.
If . . .
XZ . XY
Then . . .
m/Y . m/Z
You will prove Theorem 5-10 in Exercise 40.
X
Y
Z
G
U
I
m
C
Th
G
How do you identify
an exterior angle?
An exterior angle
must be form.
The document discusses the NP-hard Max Cut problem and provides a reduction from the NP-hard NAE-3-SAT problem to Max Cut to prove that Max Cut is also NP-hard. The reduction works by mapping clauses in a NAE-3-SAT instance to a graph instance of Max Cut, such that a solution to one problem can be translated to a solution for the other problem in polynomial time. This shows that any polynomial time algorithm for Max Cut could also be used to solve NAE-3-SAT in polynomial time. The document then provides a simple randomized approximation algorithm for Max Cut that runs in linear time.
Monte Carlo methods use random sampling to solve problems numerically. They work by setting up probabilistic models and running simulations using random numbers. This allows approximating solutions to problems in physics, finance, optimization, and other fields. Examples include estimating pi by simulating dart throws, and using a "drunken wino" random walk simulation to approximate the solution to a partial differential equation on a grid. The accuracy of Monte Carlo methods increases with more simulation iterations, requiring truly random numbers for best results.
This document contains an unsolved past paper from 2005 for the CAT exam. It includes 30 multiple choice questions across various topics like mathematics, data interpretation, and logical reasoning. The questions range in difficulty and cover topics like ratios, percentages, time/work word problems, geometry, sets and Venn diagrams, probability, and data sufficiency. The document provides the questions only - no answers or solutions are given. It is intended to allow test takers to practice solving different types of questions they may encounter on the actual exam.
A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: F...cseiitgn
The document summarizes a talk on obtaining subexponential time algorithms for NP-hard problems on planar graphs. It discusses using treewidth and tree decompositions to solve problems like 3-coloring in 2O(√n) time on n-vertex planar graphs. It also discusses the exponential time hypothesis and how it implies lower bounds, showing these algorithms are optimal up to constant factors in the exponent. The document outlines several chapters, including using grid minors and bidimensionality to obtain 2O(√k) algorithms for problems like k-path, even for some W[1]-hard problems parameterized by k.
1) The document provides a math quiz with 15 multiple choice questions covering topics in probability and statistics, including sample spaces, outcomes, experiments, events, and the fundamental counting principle.
2) It also includes 5 word scramble questions where the letters in numbers spell out terms related to exterior angles of triangles.
3) The final part of the document discusses parallel lines, transversals, and the different types of angles they form, including corresponding angles, same side interior angles, alternate interior angles, and alternate exterior angles.
I am Racheal W. I am a Probability Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, Massachusetts Institute of Technology, USA.
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- The document discusses solving a radical equation by isolating the radicals on each side of the equal sign and then squaring both sides.
- It provides an example of solving the equation 2x+4=x+7 by squaring both sides, simplifying, and finding the solution x=3.
- The final steps are to verify the solution by plugging it back into the original equation.
Area of Polygons_Composite Figures_.pptxLuisSalenga1
The document discusses finding the area of different shapes like triangles, squares, rectangles, and circles by providing the formulas and step-by-step worked examples of applying the formulas to sample figures. It also emphasizes the importance of including units when writing the result of an area calculation and integrating area concepts into other subjects.
Compiled and solved problems in geometry and trigonometry,F.SmaradanheΘανάσης Δρούγας
This document contains 29 geometry problems from Romanian textbooks for 9th and 10th grade students, compiled and solved by Florentin Smarandache. The problems cover a range of topics in geometry including properties of angles, triangles, polygons, circles, areas, and constructions. Smarandache compiled these problems during his time teaching mathematics in Romania and Morocco between 1981-1988. He provides solutions to each problem at the end of the document to serve as an educational aid for mathematics students and instructors.
Powerpoint in triangle_Inequality_Theorem.pptjawraabdul
The document discusses the triangle inequality theorem, which states that the sum of any two side lengths of a triangle must be greater than the third side length. It provides examples of determining whether a triangle can exist given three side lengths, and how to find the range of possible lengths for the third side of a triangle given the lengths of two sides.
This document contains 60 math questions from a school competition countdown round. The questions cover a variety of math topics including percentages, probability, geometry, number properties, and algebra. They range in difficulty from basic calculations to multi-step word problems.
The document discusses different types of three-dimensional shapes studied in solid geometry. It provides definitions and examples of cubes, rectangular prisms, cylinders, spheres, cones, and pyramids. It also gives the formulas for calculating the volume and surface area of these shapes. For each shape, it provides examples of applying the formulas to solve volume and surface area problems.
Finding Area of a Composite Figure (Presentation)CRISALDO CORDURA
This Presentation was adopted to Buklat-Ulat a presentation from lightning talks: Innovation. This presentation is also powered by Classpoint, one of the newest ans easiest embeded application that we can put in our presentation
Disclaimer: Some photos do not owned by the presenter and it was borrowed from google.
The document provides information about the Pythagorean theorem:
1) It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
2) It gives examples of right triangles that satisfy the theorem, such as ones with sides of 3, 4, 5 or 5, 12, 13.
3) It includes an animated proof of the theorem showing how the area of the square on the hypotenuse equals the combined areas of the squares on the other two sides.
1. The document outlines the rules for a quiz game being played between teams A-F. It details the round structure, scoring system, and rules for different rounds.
2. The last round, called the Quizzer Round, involves one team member being the quiz master who asks 5 questions to their partner in 60 seconds. The partner can have two attempts to answer each question correctly for 4 points each.
3. Hints or clues can be provided by the quiz master but they cannot read or say parts of the answers shown on the slides. Getting all 5 questions right earns a 5 point bonus for a total of 25 points at stake in the round.
1. The document outlines the rules for a quiz game being played between teams A-F. It details the round structure, scoring system, and rules for answering and passing questions.
2. The last round will be a "Quizzer Round" between the top four teams where one person will be the quiz master asking questions to their partner within a 60 second time limit.
3. The quiz master is not allowed to read full answers but can provide clues, and the partner gets two attempts per question to score 4 points each for correct answers, with a 5 point bonus for getting all questions right.
1. This document contains conceptual problems and their solutions related to measurement and vectors in physics. It covers topics like SI base units, significant figures, dimensions of physical quantities, and vector operations.
2. Conceptual questions are answered by determining the relevant concepts, relationships between quantities, and applying definitions. Numerical problems are solved by substituting values into appropriate equations.
3. Estimation problems involve approximating quantities using given values and reasonable assumptions. Diagrams are used to illustrate vector concepts and solutions.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
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Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
2. A Time Travel into Probability
Two French mathematicians:
Blaise Pascal (1623–1662)
Pierre de Fermat (1601–1665) found the
mathematics of probability in the middle of the
seventeenth century.
Their first discoveries involved the probability of
games of chance with dice and playing card s .
5. Introdoction to probability
Probability is ordinarily used to describe an attitude of
mind towards some proposition of whose truth we are not
certain.The certainty we adopt can be described in terms
of a numerical measure and this number, between 0 and 1,
we call probability.
The higher the probability of an event, the more certain
we are that the event will occur.
used widely in areas of study as mathematics, statistics,
finance, gambling, science, artificial intelligence/machine
learning and philosophy to, for example, draw inferences
about the expected frequency of events.
6. Probability of occurrence of an
event
P(n)=No of favourable outcomes
Sample Space(all possible outcomes)
7. Geometric Probability
Modern probability theory is full of formulas and
applications to modern events that are far removed from
games of chance. Instead of using algebraic formulas to
solve these probability problems, we use geometric
figures.
Geometric probability involves the distributions of length,
area, and volume for geometric objects under stated
conditions.
These geometric figures allow you to picture the
probabilities of a situation before solving the problem.
This will help you develop a sense of a reasonable solution
before you solve the problem.
9. Simple Problem Simulation
Imagine that you are throwing a dart at a 10x10 square
with a 4x4 square centrally placed. Then what is the
probability of you hitting the dart into the inner
block.?!
10. Favourable region=16 sq units
Feasible region=100 sq units
P= Area of favourable region =0.16
Feasible region
Thus you fancy a 16% chance of hitting the target!
This simple problem shows how logical is geometrical
probability compared to classical approach.
11.
12. As is evident from the previous slide if the area of the
favorable region is increased consequently the
probability of the favorable event increases .Simple
but a very important conclusion!!
13. Various models in GP
Linear Models
Area Models
Co-ordinate geometry models
3D models
14. Linear Models
Example 1:
A is driving one car in a line of cars, with about 150
feet between successive cars. Each car is 13 feet long.
At the next overpass, there is a large icicle. The icicle
is about to crash down onto the highway. If the icicle
lands on or within 30 feet of the front of a car, it will
cause an accident. What is the chance that the icicle
will cause an accident?
15.
16. The chance that the icicle will strike a passing automobile may be represented
by line segments. The sample space is the distance from the front of one car to
the front of the next car. The feasible region is
A 163 feet B the 30 feet given in the problem plus the length of the car (13
feet).
Sample space Both of these may be represented as segments
Segment AB represents the sample space.
Sample space = 150 feet (distance between cars) + 13 feet (length of a car) = 163
feet
The feasible region may be represented by segment CD.
Measure of feasible region = length of segment CD = 43 feet
18. Area Models
Example 2:The ARIZONA CRATER
A current scientific theory about the fate of dinosaurs
on the earth suggests that they were made extinct by
the effects of a large meteor which struck the earth in
the Caribbean Sea, off the coast of Central America.
The United States has been struck by large meteors in
the past, as shown by the crater in Arizona at left.
What is the chance that a meteor striking the earth
will land in the United States?
19.
20. The probability relationships for this problem may be
represented by areas.
The sample space is the surface area of the earth, or
196,940,400 square miles.
The feasible region is the area of the United States, or
3,679,245 square miles.
21. Co-ordinate Geometry Models
Another model we can use to represent probability
problems is based on geometric figures placed on a
coordinate grid. The problem in this section
demonstrates how a coordinate system may be used
to solve probability problems.
22. Example
A line segment is 8 inches long. Two points are put
on the segment at random locations. What is the
probability that the three segments formed by the two
points will make a triangle?
This problem really is tedious and cumbersome in the
classical method..See for yourself how simplified it is
now!
23.
24. What is the probability that segment AB, segment BC, and
segment CD will make a triangle?
This problem may be solved by using a coordinate system.
let the length of segment AB = x. let the length of segment
BC= y.
The length of segment CD = 8 - x - y.
The length of AB must be less than 8 inches, and the
length of BC must be less than 8 inches. This may be
represented on a coordinate system as shown in Figure.
25.
26. In addition, the sum of the two segments also must
be less than the total length of the original segment,
so AB + BC is less than 8 inches. This may be written x
+ y < 8. This inequality is shown on the coordinate
system in Figure .
27.
28. The area of triangle QRS is the sample space. All the
points in triangle QRS represent possible lengths for
segment AB and segment CD. Point K, for example,
has coordinates (6, 1). Point K represents a length of 6
for segment AB, and a length of 1 for segment Be
The relationship between the three sides of a triangle
is as follows: The sum of the lengths of any two sides
of a triangle must be longer than the length of the
third side. This results in the following inequalities,
all of which must be true:
29. x + y > 8 - x - Y (AB + BC> CD)
x + (8 - x - y) > Y (AB + CD > BC)
y + (8 - x - y) > x (BC + CD > AB)
30. The first inequality may be
simplified as follows:
x+y>8-x-y
2x + 2y > 8
x + y> 4 (the shaded area in
Figure ).
31. The second inequality may
be simplified as follows:
x + (8 - x - y) > y
8> 2y
4> y, or y < 4 (the shaded
area shown in Figure ).
32. The third inequality may be
simplified as follows:
y + (8 - x - y) > x
8> 2x
4 > x, or x < 4 (the shaded
area shown in Figure ).
33. The feasible region is the area of the
common region defined by all three
inequalities, as shown in Figure.
The probability that the three
segments will make a triangle is
found by comparing the feasible
region to the sample space:
Feasible region = Area of Triangle
EFG
Sample space =Area of Triangle QRS
34.
35. 3D models
Probability in 3D models is a problem rejoiced in
quantum physics. The normalization of the wave
function done over a 3D space is the application of
GP.
∫∫∫│w│2
dx dy dz
w-wave function
36. Buffon’s Needle Experiment
Georges Louis LeClerc, the Compte de Buffon
(1707–1788), was a mathematician who is now
famous for his needle experiment involving
probability.
Suppose we have a floor made of parallel strips
of wood, each the same width, and we drop a
needle onto the floor. What is the probability that the
needle will lie across a line between two strips?
He estimated the value of pi using this!(3.141592)
37. Bertrand’s paradox
The Bertrand paradox goes as follows: Consider an
equilateral triangle inscribed in a circle. Suppose
a chord of the circle is chosen at random. What is the
probability that the chord is longer than a side of the
triangle?
Bertrand gave three arguments, all apparently valid,
yet yielding different results.
38. The "random endpoints" method:
The probability that a random chord is
longer than a side of the inscribed
triangle is 1/3.
39. The "random midpoint" method
The probability a random chord is
longer than a side of the inscribed
triangle is 1/4.
40. The "random radius" method
The probability a random chord is
longer than a side of the inscribed
triangle is 1/2.
41. Monty Hall problem
The Monty Hall problem is a probability puzzle
loosely based on the American television game show
Let's Make a Deal and named after the show's original
host, Monty Hall.
It was originally posed in a letter by Steve Selvin to
the American Statistician in 1975 (Selvin 1975a) (Selvin
1975b). It became famous in the following form, as a
question from a reader's letter quoted in Marilyn vos
Savant's "Ask Marilyn" column in Parade magazine in
1990 (vos Savant 1990a):
42. Problem….
Suppose you're on a game show, and you're given the
choice of three doors: Behind one door is a car;
behind the others, goats. You pick a door, say No. 1,
and the host, who knows what's behind the doors,
opens another door, say No. 3, which has a goat. He
then says to you, "Do you want to pick door No. 2?" Is
it to your advantage to switch your choice?
43. In search of a new car, the player picks a door, say 1.
The game host then opens one of the other doors, say
3, to reveal a goat and offers to let the player pick door
2 instead of door 1.
44. behind door
1
behind door
2
behind door
3
result if
staying at
door #1
result if
switching to
the door
offered
Car Goat Goat Car Goat
Goat Car Goat Goat Car
Goat Goat Car Goat Car
The solution presented by vos Savant (1990b)
The probability of winning by staying with the initial choice is
therefore 1/3, while the probability of winning by switching is
2/3.
45. Player picks car
(probability 1/3)
Host reveals
either goat
Switching loses.
Player picks Goat A
(probability 1/3)
Host must
reveal Goat B
Switching wins
Player picks Goat B
(probability 1/3)
Host must
reveal Goat A
Switching wins
46. The player has an equal chance of initially selecting
the car, Goat A, or Goat B. Switching results in a win
2/3 of the time.
47. Two envelopes problem…
You have two indistinguishable envelopes that each
contain money. One contains twice as much as the
other. You may pick one envelope and keep the money
it contains. You pick at random, but before you open
the envelope, you are offered the chance to take the
other envelope instead.
48. Switching argument:
I denote by A the amount in my selected envelope.
The probability that A is the smaller amount is 1/2, and
that it is the larger amount is also 1/2.
The other envelope may contain either 2A or A/2.
If A is the smaller amount, then the other envelope
contains 2A.
If A is the larger amount, then the other envelope
contains A/2.
Thus the other envelope contains 2A with probability 1/2
and A/2 with probability 1/2.
49. So the expected value of the money in the other envelope is
This is greater than A, so I gain on average by swapping.
After the switch, I can denote that content by B and reason
in exactly the same manner as above.
I will conclude that the most rational thing to do is to swap
back again.
To be rational, I will thus end up swapping envelopes
indefinitely.
As it seems more rational to open just any envelope than to
swap indefinitely, we have a contradiction.
The puzzle: The puzzle is to find the flaw in the very
compelling line of reasoning above.
50. Common resolutionobserve that A stands for different things at different places
in the expected value calculation, step 7 above
In the first term A is the smaller amount while in the second
term A is the larger amount
To mix different instances of a variable in the same formula
like this is said to be illegitimate, so step 7 is incorrect, and
this is the cause of
By definition in one envelope is twice as much as in the
other. Denoting the lower of the two amounts by X, we
write the expected value calculation as the paradox.
Here X stands for the same thing in every term of the equation
51. we learn that 1.5X is the average expected value in either of
the envelopes, hence there is no reason to swap the
envelopes.
52. Sleeping Beauty problem…
Sleeping Beauty volunteers to undergo the following
experiment and is told all of the following details: On
Sunday she will be put to sleep. Once or twice, during the
experiment, Beauty will be wakened, interviewed, and put
back to sleep with an amnesia-inducing drug that makes her
forget that awakening. A fair coin will be tossed to
determine which experimental procedure to undertake: if
the coin comes up heads, Beauty will be wakened and
interviewed on Monday only. If the coin comes up tails, she
will be wakened and interviewed on Monday and Tuesday.
In either case, she will be wakened on Wednesday without
interview and the experiment ends.
53. Any time Sleeping Beauty is wakened and
interviewed, she is asked, "What is your belief now for
the proposition that the coin landed heads?"
54. Solutions…
Thirder position:
The thirder position argues that the probability of heads is
1/3
Halfer position:
The halfer position argues that the probability of heads is
1/2
Phenomenalist position
The phenomenalist position argues that Sleeping Beauty's
credence is meaningless until it is attached to consequences