1. The document discusses various concepts related to probability and gambling, including theoretical and empirical probability, permutations, combinations, and counting methods.
2. It provides examples of how to calculate probabilities of events using counting methods, whether order matters or not, and whether replacement is allowed or not. This includes calculating probabilities of drawing cards from a deck and rolling dice.
3. Key counting formulas are introduced, such as permutations for when order matters and replacement is allowed (P(A)= n!/(n-r)!), and combinations when order does not matter and there is no replacement (C(n,r)= n!/r!(n-r)!).
This document discusses basic probability concepts and counting methods for computing probabilities of events. It introduces probability as the chance that an uncertain event will occur between 0 and 1. It describes theoretical and empirical probabilities. It then covers counting methods for computing probabilities, distinguishing between permutations where order matters, and combinations where order does not matter. It provides examples of using permutations and combinations to calculate probabilities of events with and without replacement from populations.
1. The document discusses probability and counting methods for computing probabilities of events related to gambling and drawing cards from a deck. It introduces concepts like theoretical probability, empirical probability, permutations, combinations, and using factorials and probabilities to solve counting problems.
2. As an example, it calculates the probability of drawing a pair of the same color cards from a deck and determines it is 49% based on counting the possible combinations.
3. It also discusses using probabilities to determine rational betting strategies, like folding unless holding a pair of the same color or suit when playing against others who each drew 2 cards.
This powerpoint was used in my 7th and 8th grade classes to review the fundamental counting principle used in our probability unit. There are three independent practice problems at the end.
The document discusses probability and provides examples to explain key concepts such as sample space, probability calculations, independent and dependent events, odds, and more. Probability is defined as the chance of an event occurring and is calculated by taking the number of outcomes in the event and dividing by the total number of possible outcomes. A variety of examples using coins, cards, and dice help illustrate how to determine probabilities and odds for different scenarios.
Lecture Week 17 which hleps in study for logic andmanishhmishra001
This document summarizes a lecture on logic and problem solving that covered:
1. The basic principle of counting, permutations, and combinations. Permutations involve order while combinations do not.
2. Examples were given to demonstrate the fundamental counting principle and formulas for permutations and combinations.
3. Permutations and combinations with repetition were also discussed along with circular permutations.
4. Exercises were provided to practice applying the concepts through problems involving counting arrangements of letters, books, team selections from groups, and card selections.
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This document discusses probability and random experiments. It defines key probability concepts like sample space, events, and how to calculate probability. It provides examples of random experiments like rolling dice, flipping coins, and drawing cards. The document also includes practice problems calculating probabilities of events occurring in random experiments and their solutions.
The document discusses permutations, combinations, and probability. It provides examples and formulas for calculating the number of permutations of objects taken a certain number at a time. It also discusses combinations and provides examples. The document then provides several word problems involving permutations, combinations, and calculating probabilities of events occurring.
This document discusses basic probability concepts and counting methods for computing probabilities of events. It introduces probability as the chance that an uncertain event will occur between 0 and 1. It describes theoretical and empirical probabilities. It then covers counting methods for computing probabilities, distinguishing between permutations where order matters, and combinations where order does not matter. It provides examples of using permutations and combinations to calculate probabilities of events with and without replacement from populations.
1. The document discusses probability and counting methods for computing probabilities of events related to gambling and drawing cards from a deck. It introduces concepts like theoretical probability, empirical probability, permutations, combinations, and using factorials and probabilities to solve counting problems.
2. As an example, it calculates the probability of drawing a pair of the same color cards from a deck and determines it is 49% based on counting the possible combinations.
3. It also discusses using probabilities to determine rational betting strategies, like folding unless holding a pair of the same color or suit when playing against others who each drew 2 cards.
This powerpoint was used in my 7th and 8th grade classes to review the fundamental counting principle used in our probability unit. There are three independent practice problems at the end.
The document discusses probability and provides examples to explain key concepts such as sample space, probability calculations, independent and dependent events, odds, and more. Probability is defined as the chance of an event occurring and is calculated by taking the number of outcomes in the event and dividing by the total number of possible outcomes. A variety of examples using coins, cards, and dice help illustrate how to determine probabilities and odds for different scenarios.
Lecture Week 17 which hleps in study for logic andmanishhmishra001
This document summarizes a lecture on logic and problem solving that covered:
1. The basic principle of counting, permutations, and combinations. Permutations involve order while combinations do not.
2. Examples were given to demonstrate the fundamental counting principle and formulas for permutations and combinations.
3. Permutations and combinations with repetition were also discussed along with circular permutations.
4. Exercises were provided to practice applying the concepts through problems involving counting arrangements of letters, books, team selections from groups, and card selections.
Are complex statistics problems leaving you puzzled? Look no further! Introducing StatisticsHomeworkHelper.com, your ultimate destination for conquering statistics challenges with ease.
🔍 Unparalleled Expertise: Our team of experienced statisticians is ready to tackle any problem thrown their way. From basic concepts to advanced analyses, we've got you covered.
📈 Step-by-Step Guidance: Say goodbye to confusion! Our detailed solutions break down even the trickiest questions into manageable steps, helping you grasp the concepts along the way.
⏱️ Time-Saving Assistance: Don't waste hours struggling over a single problem. Our efficient solutions give you more time to focus on other important tasks.
🌐 Anytime, Anywhere: Access our platform 24/7 from the comfort of your home. Whether it's a late-night study session or a last-minute assignment, we're always here to help.
🎓 Excelling Made Easy: Boost your grades and gain a deeper understanding of statistics. With StatisticsHomeworkHelper.com, excelling in your studies has never been more achievable.
🚀 Try Us Today: Visit our website and experience the power of a dedicated statistics homework solver. Let's turn those daunting problems into confident victories!
📢 Spread the word and tag friends who could use a statistics study companion. Together, let's conquer statistics! 📊📚
Get ready to unravel the mysteries of statistics with ease. Visit us at StatisticsHomeworkHelper.com and never fear your homework again! 🎉
This document discusses probability and random experiments. It defines key probability concepts like sample space, events, and how to calculate probability. It provides examples of random experiments like rolling dice, flipping coins, and drawing cards. The document also includes practice problems calculating probabilities of events occurring in random experiments and their solutions.
The document discusses permutations, combinations, and probability. It provides examples and formulas for calculating the number of permutations of objects taken a certain number at a time. It also discusses combinations and provides examples. The document then provides several word problems involving permutations, combinations, and calculating probabilities of events occurring.
I am Josh U. I am a Probability Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from St. Edward’s University, USA.
I have been helping students with their homework for the past 5 years. I solve assignments related to Probability. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Probability Assignments.
1. The document provides examples and definitions for compound probability, which involves the probability of two or more independent events occurring.
2. To calculate the probability of landing on 1 and black on a spinner and choosing a black marble, the document lists the 8 possible outcomes and identifies that there are 2 outcomes where the spinner lands on 1 and the marble is black. The probability is calculated as 2/8.
3. A tree diagram is used to calculate the probability of a coin landing on heads, a spinner landing on 3, and choosing a black marble. There are 18 total outcomes and 1 outcome that meets all 3 criteria, so the probability is 1/18.
This document provides information about probability and examples of simple and compound events. It defines key probability terms like experiment, trial, outcome, sample space, and event. It explains that the probability of a simple event is the number of outcomes in the event divided by the total number of outcomes in the sample space. Compound events consist of more than one outcome. Examples of finding probabilities of simple and compound events involving coins, dice, and cards are provided. Practice problems are given to test understanding of simple vs compound events and calculating probabilities.
This document provides an overview and agenda for a statistics course. It will:
1) Recap properties of probability mass functions and finish key concepts.
2) Cover the concept of equally likely events and examples like coin flips or card draws where the probability of each outcome is equal.
3) Discuss different methods of enumeration like the multiplication principle, sampling with and without replacement, and situations where order does and does not matter.
4) Assign homework on conditional probability concepts.
There are two ways to count the number of possible outcomes of an experiment:
1) Using a tree diagram to list out all the combinations
2) Using the Fundamental Counting Principle, which involves multiplying the number of choices for each event together.
To calculate the probability of compound events (events made up of two or more simple events), you first determine if the events are independent or dependent. For independent events, the probability of one event does not affect the other event. You calculate the probability by multiplying the individual probabilities together.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 2 Solutions
Chapter 4: Probability
Probability maths lesson for education.pptWaiTengChoo
This document defines key probability terms and concepts like sample space, properties of probabilities, and how to calculate probabilities of events. It provides examples of calculating probabilities of drawing certain cards from a deck, students' music preferences in a survey, people attending a concert, and rolling dice. Probability is calculated by taking the number of favorable outcomes and dividing by the total number of possible outcomes. Events can be related or unrelated, affecting whether probabilities are multiplied or added.
The document discusses key concepts in probability theory, including sample space, events, and terminology. It provides examples of sample spaces and events for experiments like rolling dice or selecting cards from a deck. The three main approaches to assigning probabilities - classical, relative frequency, and subjective - are also outlined. The section concludes by introducing the addition rule for probabilities of unions of events.
1. The document provides examples and definitions for compound probability, which involves finding the probability of two or more sequential events occurring.
2. To calculate the probability of landing on 1 and black on a spinner and choosing a black marble, all possible outcomes are listed in a tree diagram and the probability is calculated as the number of desired outcomes over the total number of outcomes.
3. Practice problems are provided to calculate various compound probabilities involving events like coin tosses, card draws from a bag, and spinner or dice rolls.
I am Christopher, T.I am a Mathematics Assignment Expert at eduassignmenthelp.com. I hold a PhD. in Mathematics, University of Alberta, Canada. I have been helping students with their Assignments for the past 7 years. I solve assignments related to Mathematics.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com . You can also call on +1 678 648 4277 for any assistance with Mathematics Assignments.
In a family with 3 children, the probability that 2 of the children will be girls can be calculated as follows:
There are 3 children and each child can be either a boy or a girl. So there are 2 possible outcomes for each child. Using the fundamental principle of counting, there are 2 * 2 * 2 = 8 possible combinations of boys and girls. Out of these 8 combinations, 3 combinations will have exactly 2 girls. Therefore, the probability that 2 of the 3 children will be girls is 3/8.
The document discusses key concepts in probability such as trials, events, sample space, random variables, and definitions of probability. It also covers binomial and Poisson distributions and provides examples of calculating mean, variance, and probability for events following these distributions. For binomial problems, it shows how to calculate the probability of certain outcomes occurring based on the number of trials and probability of success. For Poisson problems, it demonstrates computing the probability of a given number of occurrences based on the mean rate of events.
The document contains multiple choice questions related to probability and statistics. It includes 10 probability questions related to events like drawing balls from bags, rolling dice, tossing coins, etc. It also includes 5 statistics questions related to mean, median, standard deviation, distributions, etc. The document tests the reader's understanding of key probability and statistics concepts through multiple choice questions and answers.
I am Peter C. I am a Math Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics, from London, UK. I have been helping students with their homework for the past 9 years. I solved homework related to Math.
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Probability is a measure of how likely an event is to occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, the probability of rolling a 4 on a standard 6-sided die is 1/6, as there is 1 face with a 4 and the total number of faces is 6. Probability is useful for predicting the likelihood of events but does not determine the exact outcome. Common probability concepts include sample space, sample points, events, independent and dependent events, conditional probability, and random variables.
- The document discusses key concepts in probability such as conditional probability, multiplication theorem, total probability theorem, Bayes' theorem, random variables, probability distributions, mean, variance, Bernoulli trials, and binomial distribution.
- It provides examples and formulas for calculating probabilities of events. It also lists very short answer type, short answer type, and long answer type questions related to probability with answers.
I am Arcady N. I am a Maths Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, Queen’s University. I have been helping students with their assignments for the past 11 years. I solve assignments related to Maths.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Maths.
This document provides an overview of a probability and statistics course, including the grading criteria, topics that will be covered like machine learning, probability in real life examples, key terminology, types of events, and how probability is used in programming. The course will cover sample space, events, counting sample points, and probability of an event. Assignments will make up 20% of the grade, with the midterm and final exam making up 30% and 40% respectively. Topics that will be discussed include random variables, empirical vs theoretical probability, independent and mutually exclusive events, and probability distributions. Examples are provided for calculating probabilities of different events.
This document provides a summary of key concepts related to normal distribution for statistical analysis, including definitions of frequency distribution and important probability distributions like normal, binomial, chi square, student T, and F distribution. It also lists the assumptions of normal and binomial distributions and provides examples of probability calculations for scenarios involving dice, cards, samples, and more.
This document provides a summary of key concepts related to normal distribution for statistical analysis, including definitions of frequency distribution and important probability distributions like normal, binomial, chi square, student T, and F distribution. It also lists the assumptions of normal and binomial distributions and provides examples of probability calculations for scenarios involving dice, cards, samples, and more.
Quarter III: Permutation and CombinationIzah Catli
The document provides examples of combination problems involving selecting items from sets with certain constraints. The first example asks the reader to calculate the number of possible 4-digit PIN combinations using the digits 7, 3, 5, and 4 where the last digit is 4. The second example asks students to list the possible combinations of visiting 5 tourist attractions in Tagaytay with no more than 3 places per combination. The third example asks the reader to calculate the number of combinations for a 9-piece tart box allowing up to 3 different flavors.
Math 9 : Quarter III Similarity : Ratio and ProportionIzah Catli
The document discusses the concepts of ratio, proportion, and congruency similarity in geometry. It defines ratio as a way to compare quantities and proportion as the equality of two ratios. Examples are provided for writing ratios as fractions or using colons, and for solving proportions using cross multiplication or setting the product of extremes equal to the product of means.
I am Josh U. I am a Probability Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from St. Edward’s University, USA.
I have been helping students with their homework for the past 5 years. I solve assignments related to Probability. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Probability Assignments.
1. The document provides examples and definitions for compound probability, which involves the probability of two or more independent events occurring.
2. To calculate the probability of landing on 1 and black on a spinner and choosing a black marble, the document lists the 8 possible outcomes and identifies that there are 2 outcomes where the spinner lands on 1 and the marble is black. The probability is calculated as 2/8.
3. A tree diagram is used to calculate the probability of a coin landing on heads, a spinner landing on 3, and choosing a black marble. There are 18 total outcomes and 1 outcome that meets all 3 criteria, so the probability is 1/18.
This document provides information about probability and examples of simple and compound events. It defines key probability terms like experiment, trial, outcome, sample space, and event. It explains that the probability of a simple event is the number of outcomes in the event divided by the total number of outcomes in the sample space. Compound events consist of more than one outcome. Examples of finding probabilities of simple and compound events involving coins, dice, and cards are provided. Practice problems are given to test understanding of simple vs compound events and calculating probabilities.
This document provides an overview and agenda for a statistics course. It will:
1) Recap properties of probability mass functions and finish key concepts.
2) Cover the concept of equally likely events and examples like coin flips or card draws where the probability of each outcome is equal.
3) Discuss different methods of enumeration like the multiplication principle, sampling with and without replacement, and situations where order does and does not matter.
4) Assign homework on conditional probability concepts.
There are two ways to count the number of possible outcomes of an experiment:
1) Using a tree diagram to list out all the combinations
2) Using the Fundamental Counting Principle, which involves multiplying the number of choices for each event together.
To calculate the probability of compound events (events made up of two or more simple events), you first determine if the events are independent or dependent. For independent events, the probability of one event does not affect the other event. You calculate the probability by multiplying the individual probabilities together.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 2 Solutions
Chapter 4: Probability
Probability maths lesson for education.pptWaiTengChoo
This document defines key probability terms and concepts like sample space, properties of probabilities, and how to calculate probabilities of events. It provides examples of calculating probabilities of drawing certain cards from a deck, students' music preferences in a survey, people attending a concert, and rolling dice. Probability is calculated by taking the number of favorable outcomes and dividing by the total number of possible outcomes. Events can be related or unrelated, affecting whether probabilities are multiplied or added.
The document discusses key concepts in probability theory, including sample space, events, and terminology. It provides examples of sample spaces and events for experiments like rolling dice or selecting cards from a deck. The three main approaches to assigning probabilities - classical, relative frequency, and subjective - are also outlined. The section concludes by introducing the addition rule for probabilities of unions of events.
1. The document provides examples and definitions for compound probability, which involves finding the probability of two or more sequential events occurring.
2. To calculate the probability of landing on 1 and black on a spinner and choosing a black marble, all possible outcomes are listed in a tree diagram and the probability is calculated as the number of desired outcomes over the total number of outcomes.
3. Practice problems are provided to calculate various compound probabilities involving events like coin tosses, card draws from a bag, and spinner or dice rolls.
I am Christopher, T.I am a Mathematics Assignment Expert at eduassignmenthelp.com. I hold a PhD. in Mathematics, University of Alberta, Canada. I have been helping students with their Assignments for the past 7 years. I solve assignments related to Mathematics.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com . You can also call on +1 678 648 4277 for any assistance with Mathematics Assignments.
In a family with 3 children, the probability that 2 of the children will be girls can be calculated as follows:
There are 3 children and each child can be either a boy or a girl. So there are 2 possible outcomes for each child. Using the fundamental principle of counting, there are 2 * 2 * 2 = 8 possible combinations of boys and girls. Out of these 8 combinations, 3 combinations will have exactly 2 girls. Therefore, the probability that 2 of the 3 children will be girls is 3/8.
The document discusses key concepts in probability such as trials, events, sample space, random variables, and definitions of probability. It also covers binomial and Poisson distributions and provides examples of calculating mean, variance, and probability for events following these distributions. For binomial problems, it shows how to calculate the probability of certain outcomes occurring based on the number of trials and probability of success. For Poisson problems, it demonstrates computing the probability of a given number of occurrences based on the mean rate of events.
The document contains multiple choice questions related to probability and statistics. It includes 10 probability questions related to events like drawing balls from bags, rolling dice, tossing coins, etc. It also includes 5 statistics questions related to mean, median, standard deviation, distributions, etc. The document tests the reader's understanding of key probability and statistics concepts through multiple choice questions and answers.
I am Peter C. I am a Math Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics, from London, UK. I have been helping students with their homework for the past 9 years. I solved homework related to Math.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com. You can also call on +1 678 648 4277 for any assistance with Math Homework.
Probability is a measure of how likely an event is to occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, the probability of rolling a 4 on a standard 6-sided die is 1/6, as there is 1 face with a 4 and the total number of faces is 6. Probability is useful for predicting the likelihood of events but does not determine the exact outcome. Common probability concepts include sample space, sample points, events, independent and dependent events, conditional probability, and random variables.
- The document discusses key concepts in probability such as conditional probability, multiplication theorem, total probability theorem, Bayes' theorem, random variables, probability distributions, mean, variance, Bernoulli trials, and binomial distribution.
- It provides examples and formulas for calculating probabilities of events. It also lists very short answer type, short answer type, and long answer type questions related to probability with answers.
I am Arcady N. I am a Maths Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, Queen’s University. I have been helping students with their assignments for the past 11 years. I solve assignments related to Maths.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Maths.
This document provides an overview of a probability and statistics course, including the grading criteria, topics that will be covered like machine learning, probability in real life examples, key terminology, types of events, and how probability is used in programming. The course will cover sample space, events, counting sample points, and probability of an event. Assignments will make up 20% of the grade, with the midterm and final exam making up 30% and 40% respectively. Topics that will be discussed include random variables, empirical vs theoretical probability, independent and mutually exclusive events, and probability distributions. Examples are provided for calculating probabilities of different events.
This document provides a summary of key concepts related to normal distribution for statistical analysis, including definitions of frequency distribution and important probability distributions like normal, binomial, chi square, student T, and F distribution. It also lists the assumptions of normal and binomial distributions and provides examples of probability calculations for scenarios involving dice, cards, samples, and more.
This document provides a summary of key concepts related to normal distribution for statistical analysis, including definitions of frequency distribution and important probability distributions like normal, binomial, chi square, student T, and F distribution. It also lists the assumptions of normal and binomial distributions and provides examples of probability calculations for scenarios involving dice, cards, samples, and more.
Quarter III: Permutation and CombinationIzah Catli
The document provides examples of combination problems involving selecting items from sets with certain constraints. The first example asks the reader to calculate the number of possible 4-digit PIN combinations using the digits 7, 3, 5, and 4 where the last digit is 4. The second example asks students to list the possible combinations of visiting 5 tourist attractions in Tagaytay with no more than 3 places per combination. The third example asks the reader to calculate the number of combinations for a 9-piece tart box allowing up to 3 different flavors.
Math 9 : Quarter III Similarity : Ratio and ProportionIzah Catli
The document discusses the concepts of ratio, proportion, and congruency similarity in geometry. It defines ratio as a way to compare quantities and proportion as the equality of two ratios. Examples are provided for writing ratios as fractions or using colons, and for solving proportions using cross multiplication or setting the product of extremes equal to the product of means.
This document discusses identifying and addressing bias and prejudice. It begins by outlining objectives of detecting and showing examples of bias and prejudice, performing related tasks, and expressing personal thoughts on the topics. It then describes a game where players identify choices among people with different traits. It provides potential examples of bias and prejudice based on characteristics like gender, religion, socioeconomic status. It defines bias as unfair favoritism and prejudice as unfavorable pre-judging. Students are asked questions about labeling and judging people and shown examples to analyze. They are assigned a group activity to demonstrate understanding of biases and prejudices in professions, personal experiences, and comics.
This document discusses different types of permutations with restrictions, including linear permutations where nPr = n!, permutations of n objects taken r at a time, circular permutations, permutations where items must stay together, and permutations where items cannot stay together. It also mentions permutations with repetition.
The parish has launched an online Catholic Bible study called Aral-PanDASAL. It will be livestreamed on the parish's Facebook page every Friday from 8-9pm. Parishioners are asked to follow the Facebook page to receive notifications about parish activities. The document also includes announcements about Masses and various prayers and hymns.
The document discusses the concept of joint variation and provides examples of situations that illustrate joint variation, such as how the volume of a cylinder varies jointly based on its height and the square of the radius. Readers are given problems to solve involving joint variation and concepts are explained, such as how changing one quantity in a jointly varying relationship will affect the other quantity. Various examples are used to help conceptualize joint variation.
This document provides information and examples about direct variation, including:
- Definitions of direct variation and how it relates to the dependent and independent variables.
- Examples of direct variation relationships shown in tables with the constant of variation calculated.
- Word problems demonstrating how to use direct variation to find unknown values. Both setting up proportions and using the direct variation equation y=kx are presented.
- An activity is assigned for students to practice direct variation concepts by solving problems and writing a narrative about how it has helped them.
The document discusses the eight laws of exponents:
1) Exponential form indicates how many times the base is multiplied by itself using the exponent.
2) When multiplying powers with the same base, add the exponents.
3) When dividing powers with the same base, subtract the exponents.
4) When raising a power to another power, multiply the exponents.
5) When taking a power of a product, apply the exponent to each factor.
6) When taking a power of a quotient, apply the exponent to the numerator and denominator.
7) A negative exponent changes the base to its reciprocal with a positive exponent.
8) Any base raised to the zero power equals one.
This document defines various circle terms like tangent, secant, concentric circles, and common tangents. It then discusses theorems for finding angle measures based on whether the vertex is on the circle, inside the circle, or outside the circle. For a vertex on the circle, the angle measure is half the intercepted arc. For inside, it's half the sum of intercepted arcs. For outside, it's half the difference of intercepted arcs. Examples are provided to demonstrate applying these theorems to find unknown angle measures.
The document discusses exponents and laws regarding operations with exponents. It defines exponents as indicating the number of times a base is multiplied by itself. The laws covered include: keeping the base and adding exponents when multiplying powers; keeping the base and subtracting exponents when dividing powers; multiplying exponents when raising a power to an exponent; applying the exponent to all factors when taking a power of a product or quotient; changing the base to its reciprocal and making the exponent positive when the exponent is negative; and any base with a zero exponent equals one. Examples are provided to demonstrate how to apply each law.
This document defines exponents and how they are used to represent repeated multiplication of a base number. Exponents tell how many times the base number is multiplied by itself. The exponent stands for the number of times the base is multiplied. Examples are provided of how to read exponents aloud and write them in standard form by multiplying out the base the number of times indicated by the exponent.
The document discusses the geometric mean of numbers and the properties of right triangles when an altitude is drawn to the hypotenuse. It states that the altitude is the geometric mean between the hypotenuse segments and that each leg is the geometric mean between the hypotenuse and adjacent hypotenuse segment. It also provides an example problem about finding distances between a person and objects in a room.
This document outlines 8 circle theorems: 1) The angle at the center is twice the angle at the circumference. 2) The angle in a semi-circle is 90 degrees. 3) Angles in the same segment are equal. 4) Opposite angles in a cyclic quadrilateral add up to 180 degrees. 5) The lengths of the two tangents from a point to a circle are equal. 6) The angle between a tangent and a radius in a circle is 90 degrees. 7) The angle between the tangent and chord at the point of contact equals the angle in the alternate segment. 8) A perpendicular from the center bisects a chord.
The document discusses the discriminant and how it is used to determine the number of solutions to a quadratic equation. It defines the discriminant as b^2 - 4ac and explains that:
- If the discriminant is positive, there are 2 solutions
- If the discriminant is 0, there is 1 solution
- If the discriminant is negative, there are no real solutions.
Several examples are provided to demonstrate calculating the discriminant and using it to determine the nature of the roots. The key aspects are to identify the coefficients a, b, and c and then evaluate and compare b^2 - 4ac to determine how many solutions exist.
The document defines key terms related to circles such as radius, diameter, chord, arc, semicircle, and segment. It explains that a circle is a closed curve where all points are equidistant from the center. A radius connects the center to any point on the circle, while a diameter connects two points on the circle and passes through the center. An arc is part of the circumference between two points on the circle. A semicircle is half of a circle, and a chord connects any two points on the circle. The document also provides examples of circles in daily life, music, and sports.
This document discusses proportionality theorems and special right triangle theorems. It then provides 3 learning tasks involving solving word problems about special right triangles, asking for lengths of sides and heights in simplest radical form. The first problem involves a 45-45-90 right triangle with a base of 6 ft, asking for the height and hypotenuse. The second problem involves finding the height of an equilateral triangle with a side length of 34 inches.
This document discusses arithmetic sequences. It defines key terms like sequence, term, and common difference. It explains how to identify if a set of numbers forms an arithmetic sequence based on having a constant difference between terms. Methods are provided for finding the next term, the nth term, and representing the sequence as a linear function. Examples demonstrate how to apply these concepts to solve problems involving arithmetic sequences.
The document discusses measuring positions such as quartiles, deciles, and percentiles in grouped data. It provides steps for calculating each measure of position and interpreting the results. Examples are given to find the second quartile, fifth decile, and 30th percentile in a Scrabble competition. Additionally, the document discusses analyzing energy expenditure data from a Zumba class and grade data from a survey of 10 students.
The document discusses probability and events, defining key terms like experiment, outcome, sample space, and event. It provides examples of simple and compound events, and explains how to calculate the probability of simple events using the formula of number of outcomes in the event over the total number of possible outcomes. Rules for probability are also outlined, such as the probability of any event being between 0 and 1 and the sum of probabilities of all outcomes equaling 1.
CapTechTalks Webinar Slides June 2024 Donovan Wright.pptxCapitolTechU
Slides from a Capitol Technology University webinar held June 20, 2024. The webinar featured Dr. Donovan Wright, presenting on the Department of Defense Digital Transformation.
How to Setup Default Value for a Field in Odoo 17Celine George
In Odoo, we can set a default value for a field during the creation of a record for a model. We have many methods in odoo for setting a default value to the field.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
🔥🔥🔥🔥🔥🔥🔥🔥🔥
إضغ بين إيديكم من أقوى الملازم التي صممتها
ملزمة تشريح الجهاز الهيكلي (نظري 3)
💀💀💀💀💀💀💀💀💀💀
تتميز هذهِ الملزمة بعِدة مُميزات :
1- مُترجمة ترجمة تُناسب جميع المستويات
2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
🔥🔥🔥🔥🔥🔥🔥🔥🔥
How to Download & Install Module From the Odoo App Store in Odoo 17Celine George
Custom modules offer the flexibility to extend Odoo's capabilities, address unique requirements, and optimize workflows to align seamlessly with your organization's processes. By leveraging custom modules, businesses can unlock greater efficiency, productivity, and innovation, empowering them to stay competitive in today's dynamic market landscape. In this tutorial, we'll guide you step by step on how to easily download and install modules from the Odoo App Store.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
2. A gambling experiment
Everyone in the room takes 2 cards
from the deck (keep face down)
Rules, most to least valuable:
Pair of the same color (both red or both black)
Mixed-color pair (1 red, 1 black)
Any two cards of the same suit
Any two cards of the same color
In the event of a tie, highest card wins (ace is top)
3. What do you want to bet?
Look at your two cards.
Will you fold or bet?
What is the most rational strategy given
your hand?
4. Rational strategy
There are N people in the room
What are the chances that someone in
the room has a better hand than you?
Need to know the probabilities of
different scenarios
We’ll return to this later in the lecture…
5. Probability
Probability – the chance that an uncertain
event will occur (always between 0 and 1)
Symbols:
P(event A) = “the probability that event A will occur”
P(red card) = “the probability of a red card”
P(~event A) = “the probability of NOT getting event A” [complement]
P(~red card) = “the probability of NOT getting a red card”
P(A & B) = “the probability that both A and B happen” [joint probability]
P(red card & ace) = “the probability of getting a red ace”
6. Assessing Probability
1. Theoretical/Classical probability—based on theory (a
priori understanding of a phenomena)
e.g.: theoretical probability of rolling a 2 on a standard die is 1/6
theoretical probability of choosing an ace from a standard deck
is 4/52
theoretical probability of getting heads on a regular coin is 1/2
2. Empirical probability—based on empirical data
e.g.: you toss an irregular die (probabilities unknown) 100 times and
find that you get a 2 twenty-five times; empirical probability of
rolling a 2 is 1/4
empirical probability of an Earthquake in Bay Area by 2032 is .62
(based on historical data)
empirical probability of a lifetime smoker developing lung cancer
is 15 percent (based on empirical data)
7. Recent headlines on earthquake
probabiilites…
http://www.guardian.co.uk/world/2011/may/26/italy-
quake-experts-manslaughter-charge
8. Computing theoretical
probabilities:counting methods
Great for gambling! Fun to compute!
If outcomes are equally likely to occur…
outcomes
of
#
total
occur
can
A
ways
of
#
)
(
A
P
Note: these are called “counting methods” because we have
to count the number of ways A can occur and the number
of total possible outcomes.
9. Counting methods: Example 1
0769
.
52
4
deck
in the
cards
of
#
deck
in the
aces
of
#
)
ace
an
draw
(
P
Example 1: You draw one card from a deck of
cards. What’s the probability that you draw an ace?
10. Counting methods: Example 2
Example 2. What’s the probability that you draw 2 aces when you draw
two cards from the deck?
52
4
deck
in the
cards
of
#
deck
in the
aces
of
#
)
draw
first
on
ace
draw
(
P
51
3
deck
in the
cards
of
#
deck
in the
aces
of
#
)
too
draw
second
on
ace
an
draw
(
P
51
3
x
52
4
ace)
AND
ace
draw
(
P
This is a “joint probability”—we’ll get back to this on Wednesday
11. Counting methods: Example 2
Numerator: AA, AA, AA, AA, AA, AA, AA, AA, AA,
AA, AA, or AA = 12
draw
could
you
sequences
card
-
2
different
of
#
ace
ace,
draw
can
you
ways
of
#
)
aces
2
draw
(
P
.
.
.
52 cards 51 cards
.
.
.
Two counting method ways to calculate this:
1. Consider order:
Denominator = 52x51 = 2652 -- why?
51
52
12
)
aces
2
draw
(
x
P
13. Summary of Counting Methods
Counting methods for computing probabilities
With replacement
Without replacement
Permutations—
order matters!
Combinations—
Order doesn’t
matter
Without replacement
14. Summary of Counting Methods
Counting methods for computing probabilities
With replacement
Without replacement
Permutations—
order matters!
15. Permutations—Order matters!
A permutation is an ordered arrangement of objects.
With replacement=once an event occurs, it can occur again
(after you roll a 6, you can roll a 6 again on the same die).
Without replacement=an event cannot repeat (after you draw
an ace of spades out of a deck, there is 0 probability of
getting it again).
16. Summary of Counting Methods
Counting methods for computing probabilities
With replacement
Permutations—
order matters!
17. With Replacement – Think coin tosses, dice, and DNA.
“memoryless” – After you get heads, you have an equally likely chance of getting a
heads on the next toss (unlike in cards example, where you can’t draw the same card
twice from a single deck).
What’s the probability of getting two heads in a row (“HH”) when tossing a coin?
H
H
T
T
H
T
Toss 1:
2 outcomes
Toss 2:
2 outcomes 22 total possible outcomes: {HH, HT, TH, TT}
Permutations—with replacement
outcomes
possible
2
HH
get
way to
1
)
( 2
HH
P
18. What’s the probability of 3 heads in a row?
outcomes
possible
8
2
1
)
( 3
HHH
P
Permutations—with replacement
H
H
T
T
H
T
Toss 1:
2 outcomes
Toss 2:
2 outcomes
Toss 3:
2 outcomes
H
T
H
T
H
T
H
T
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
19. 36
1
6
6
6,
roll
way to
1
)
6
,
6
( 2
P
When you roll a pair of dice (or 1 die twice),
what’s the probability of rolling 2 sixes?
What’s the probability of rolling a 5 and a 6?
36
2
6
6,5
or
5,6
:
ways
2
)
6
&
5
( 2
P
Permutations—with replacement
20. Summary: order matters, with
replacement
Formally, “order matters” and “with
replacement” use powers
r
events
of
#
the
n
event)
per
outcomes
possible
(#
21. Summary of Counting Methods
Counting methods for computing probabilities
Without replacement
Permutations—
order matters!
22. Permutations—without
replacement
Without replacement—Think cards (w/o
reshuffling) and seating arrangements.
Example: You are moderating a debate of
gubernatorial candidates. How many different
ways can you seat the panelists in a row? Call
them Arianna, Buster, Camejo, Donald, and Eve.
23. Permutation—without
replacement
“Trial and error” method:
Systematically write out all combinations:
A B C D E
A B C E D
A B D C E
A B D E C
A B E C D
A B E D C
.
.
.
Quickly becomes a pain!
Easier to figure out patterns using a the
probability tree!
25. Permutation—without
replacement
What if you had to arrange 5 people in only 3 chairs
(meaning 2 are out)?
!
2
!
5
1
2
1
2
3
4
5
x
x
x
x
x
E
B
A
C
D
E
A
B
D
A
B
C
D
Seat One:
5 possible
Seat Two:
Only 4 possible
E
B
D
Seat Three:
only 3 possible
)!
3
5
(
!
5
3
4
5 x
x
28. Summary: order matters,
without replacement
Formally, “order matters” and “without
replacement” use factorials
)
1
)...(
2
)(
1
(
or
)!
(
!
draws)!
or
chairs
cards
or
people
(
cards)!
or
people
(
r
n
n
n
n
r
n
n
r
n
n
29. Practice problems:
1. A wine taster claims that she can distinguish
four vintages or a particular Cabernet. What
is the probability that she can do this by
merely guessing (she is confronted with 4
unlabeled glasses)? (hint: without
replacement)
2. In some states, license plates have six
characters: three letters followed by three
numbers. How many distinct such plates are
possible? (hint: with replacement)
30. Answer 1
1. A wine taster claims that she can distinguish four vintages or a particular
Cabernet. What is the probability that she can do this by merely
guessing (she is confronted with 4 unlabeled glasses)? (hint: without
replacement)
P(success) = 1 (there’s only way to get it right!) / total # of guesses she could make
Total # of guesses one could make randomly:
glass one: glass two: glass three: glass four:
4 choices 3 vintages left 2 left no “degrees of freedom” left
P(success) = 1 / 4! = 1/24 = .04167
= 4 x 3 x 2 x 1 = 4!
31. Answer 2
2. In some states, license plates have six characters: three letters
followed by three numbers. How many distinct such plates are
possible? (hint: with replacement)
263 different ways to choose the letters and 103 different ways to
choose the digits
total number = 263 x 103 = 17,576 x 1000 = 17,576,000
32. Summary of Counting Methods
Counting methods for computing probabilities
Combinations—
Order doesn’t
matter
Without replacement
34. Combinations
2
)!
2
52
(
!
52
2
51
52
x
How many two-card hands can I draw from a deck when order
does not matter (e.g., ace of spades followed by ten of clubs is
the same as ten of clubs followed by ace of spades)
.
.
.
52 cards 51 cards
.
.
.
35. Combinations
?
48
49
50
51
52 x
x
x
x
How many five-card hands can I draw from a deck when order
does not matter?
.
.
.
52 cards
51 cards
.
.
.
.
.
.
.
.
.
.
.
.
50 cards
49 cards
48 cards
39. Combinations
How many unique 2-card sets out of 52
cards?
5-card sets?
r-card sets?
r-card sets out of n-cards?
!
2
)!
2
52
(
!
52
2
51
52
x
!
5
)!
5
52
(
!
52
!
5
48
49
50
51
52
x
x
x
x
!
)!
52
(
!
52
r
r
!
)!
(
!
r
r
n
n
n
r
40. Summary: combinations
If r objects are taken from a set of n objects without replacement and disregarding
order, how many different samples are possible?
Formally, “order doesn’t matter” and “without replacement”
use choosing
!
)!
(
!
r
r
n
n
n
r
41. Examples—Combinations
A lottery works by picking 6 numbers from 1 to 49.
How many combinations of 6 numbers could you
choose?
816
,
983
,
13
!
6
!
43
!
49
49
6
Which of course means that your probability of winning is 1/13,983,816!
42. Examples
How many ways can you get 3 heads in 5 coin tosses?
10
!
2
!
3
!
5
5
3
43. Summary of Counting
Methods
Counting methods for computing probabilities
With replacement: nr
Permutations—
order matters!
Without replacement:
n(n-1)(n-2)…(n-r+1)=
Combinations—
Order doesn’t
matter
Without
replacement:
)!
(
!
r
n
n
!
)!
(
!
r
r
n
n
n
r
44. Gambling, revisited
What are the probabilities of the
following hands?
Pair of the same color
Pair of different colors
Any two cards of the same suit
Any two cards of the same color
45. Pair of the same color?
P(pair of the same color) =
ns
combinatio
card
two
of
#
total
color
same
of
pairs
#
Numerator = red aces, black aces; red kings, black kings;
etc.…= 2x13 = 26
1326
2
52x51
r
Denominato 2
52
C
chance
1.96%
1326
26
color)
same
the
of
P(pair
So,
46. Any old pair?
P(any pair) =
1326
ns
combinatio
card
two
of
#
total
pairs
#
chance
5.9%
1326
78
pair)
P(any
pairs
possible
total
78
13x6
...
6
2
3
4
!
2
!
2
4!
C
kings
of
pairs
possible
different
of
number
6
2
3
4
!
2
!
2
4!
C
aces
of
pairs
possible
different
of
number
2
4
2
4
x
x
47. Two cards of same suit?
312
4
78
4
11!2!
13!
suits
4
C
:
Numerator 2
13
x
x
x
chance
23.5%
1326
312
suit)
same
the
of
cards
P(two
48. Two cards of same color?
Numerator: 26C2 x 2 colors = 26!/(24!2!) = 325 x 2 = 650
Denominator = 1326
So, P (two cards of the same color) = 650/1326 = 49% chance
A little non-intuitive? Here’s another way to look at it…
.
.
.
52 cards
26 red branches
26 black branches
From a Red branch: 26 black left, 25 red left
.
.
.
From a Black branch: 26 red left, 25 black left
26x25 RR
26x26 RB
26x26 BR
26x25 BB
50/102
Not
quite
50/100
49. Rational strategy?
To bet or fold?
It would be really complicated to take into
account the dependence between hands in the
class (since we all drew from the same deck), so
we’re going to fudge this and pretend that
everyone had equal probabilities of each type of
hand (pretend we have “independence”)…
Just to get a rough idea...
50. Rational strategy?
**Trick! P(at least 1) = 1- P(0)
P(at least one same-color pair in the class)=
1-P(no same-color pairs in the whole class)=
pair
color
-
same
one
least
at
of
chance
.4%
55
.446
-
1
(.98)
-
1 40
40
)
98
(.
)....
98
(.
*
)
98
(.
*
)
98
(.
class)
whole
in the
pairs
color
-
same
P(no
.98
.0196
-
1
pair)
color
-
same
a
get
t
don'
P(I
51. Rational strategy?
P(at least one pair)= 1-P(no pairs)=
1-(.94)40=1-8%=92% chance
P(>=1 same suit)= 1-P(all different suits)=
1-(.765)40=1-.00002 ~ 100%
P(>=1 same color) = 1-P(all different colors)=
1-(.51) 40=1-.000000000002 ~ 100%
52. Rational strategy…
Fold unless you have a same-color pair or a
numerically high pair (e.g., Queen, King,
Ace).
How does this compare to class?
-anyone with a same-color pair?
-any pair?
-same suit?
-same color?
53. Practice problem:
A classic problem: “The Birthday Problem.” What’s
the probability that two people in a class of 25 have
the same birthday? (disregard leap years)
What would you guess is the probability?
54. Birthday Problem Answer
1. A classic problem: “The Birthday Problem.” What’s the
probability that two people in a class of 25 have the same
birthday? (disregard leap years)
**Trick! 1- P(none) = P(at least one)
Use complement to calculate answer. It’s easier to calculate 1- P(no
matches) = the probability that at least one pair of people have
the same birthday.
What’s the probability of no matches?
Denominator: how many sets of 25 birthdays are there?
--with replacement (order matters)
36525
Numerator: how many different ways can you distribute 365 birthdays
to 25 people without replacement?
--order matters, without replacement:
[365!/(365-25)!]= [365 x 364 x 363 x 364 x ….. (365-24)]
P(no matches) = [365 x 364 x 363 x 364 x ….. (365-24)] / 36525
55. Use SAS as a calculator
Use SAS as calculator… (my calculator won’t do factorials as high as 365, so I had to
improvise by using a loop…which you’ll learn later in HRP 223):
%LET num = 25; *set number in the class;
data null;
top=1; *initialize numerator;
do j=0 to (&num-1) by 1;
top=(365-j)*top;
end;
BDayProb=1-(top/365**&num);
put BDayProb;
run;
From SAS log:
0.568699704, so 57% chance!
56. For class of 40 (our class)?
For class of 40?
10 %LET num = 40; *set number in the class;
11 data null;
12 top=1; *initialize numerator;
13 do j=0 to (&num-1) by 1;
14 top=(365-j)*top;
15 end;
16 BDayProb=1-(top/365**&num);
17 put BDayProb;
18 run;
0.891231809, i.e. 89% chance of a
match!