The subject of probability theory is the foundation upon which all of statistics is built, providing a means for modeling populations, experiments, or almost anything else that could be considered a random phenomenon. P (Getting numbers greater than 2) = 6/8 = 3/4. Clue #3. , P (A) = n (A)/n (S). This section provides the lecture notes for each session of the course. Number of odd numbers = 4. An event is a subset of a sample space associated with a Feb 4, 2021 · This entry was posted in Probability and tagged 0580, exercise, IGCSE MATHS, NOTES, practice, Probability, REVISION by Suresh Goel. Solution. (iii) Probability that the arrow will point at the odd numbers: Odd number of outcomes = 1, 3, 5, 7. Modes of Convergence in Probability Theory David Mandel November 5, 2015 Below, x a probability space (;F;P) on which all random variables fX ng and X are de ned. It can range from $0$ to $1$ a positive value. 5 minutes) for the elevator to arrive, then we calculate the following probability using the pdf and the fourth property in Definition 4. probability function of X is defined as fX(x) = P(X = x) Endpoints of intervals. Prof. probability mass function (p 0;p 1;:::), you simply need to compute the sum P 1 i=1 p i. Draw a spinner diagram to match the clues decimal, and percent, in the table below. (28/50 = 0. Set books The notes cover only material in the Probability I course. 273 kB Introduction to Probability, Selected Textbook Summary Material Download File . Jan 9, 2024 · CBSE Class 12 Maths Notes Chapter 13 Probability Event: A subset of the sample space associated with a random experiment is called an event or a case. Perform an experiment and collect data on a Preface This is a very slight revision of the notes used for Math 19b in the Spring 2009 semester. Some Basic Definitions. Topic 2. 10) These conditions are necessary for a measure to be a valid probability measure. The spinner is divided into eight equal sections, each colored red, blue, or yellow. Jan 22, 2024 · The probability of an event E is a number P (E) such that: 0 ≤ P (E) ≤ 1. c. There are 3 ∗ 2 ∗ 1 = 3! arrangements of 3 objects. 1 - Basic Definitions of Probability. The videos in Part I introduce the general framework of probability models, multiple discrete or continuous random variables, expectations, conditional distributions, and various powerful tools of general applicability. a x f(x) 1 34 3. search; pdf. These notes will certainly provide the simplest explanation of the mathematical concepts of Probability and help you learn and practice the sums more efficiently. It presents a thorough treatment of probability ideas andtechniques necessary for a form understanding of the subject. For example, suppose that the probability of having lung cancer is P(C) = 0:001 and that the probability of being a smoker is P(SM) = 0:25. Addition Rule for OR Events: P(A or B) = P(A) + P(B) – P(A and B) g. Concise resources for the CIE A Level Maths: Probability & Statistics 1course. Guided notes or flippables are provided for each lesson as well as an activity and a practice sheet. Engineering Mathematics Chapter 5 Probability & Statistics For GATE CSE probability. Statements *and proofs* of the maximal ergodic theorem and Birkho ’s almost everywhere er- Probability Notes - Free download as Word Doc (. ( 26/50 = 0. De- 1. NCERT Solutions for Class 10 Maths Chapter 15 Probability. Create organized lists, tables, tree diagrams, and simulations to determine the probability of compound events. Moment generating functions 93 13. 5 Moments and variance 3. 3 days ago · Download Probability JEE Advanced Notes Free PDF. Part I: The Fundamentals. Propositions marked with \F" denote results that rely on our nite measure space. eSaral provides a series of detailed chapter wise notes for all the Subjects of class 11th and 12th. A sample space is the set of all possible outcomes of an experiment. The text can be usedin a variety of Probability PDF notes are available on our website so that the student can get the benefits of the notes and get good marks in the exam. 9 Poisson random variables 2/13 These class notes are the currently used textbook for “Probabilistic Systems Analysis,” an introductory probability course at the Massachusetts Institute of Technology. 11. Spring 2020. Through this it is also useful for the students who think that Probability is a tough chapter, it also makes them comfortable to understand every concept or theorem in detail. For example, if X takes values 0 1 2. 2. You need at most one of the three textbooks listed below, but you will need the statistical tables. Best of Luck. The rst basic notion in probability theory is that of sample space. P (Getting odd numbers) = 4/8 = ½. ity of the elements of our sample space is 1. 175. An experiment whose outcomes cannot be predicted or determined in advance is called a random experiment. Coariancev and Correlations. Theory of probability and Stochastic Processes-Pradip Kumar Gosh, University Press 2. I have attempted to lay a sound foundation in the theory of probability to support the basics of statistical inference presented in Lectures 11 and 12. Find the probability of not getting a 3 or 5 while throwing a die. These lecture notes are a nece. It defines probability as the long-run relative frequency of an event occurring as the number of trials approaches infinity. 1 Statistical Measures. D. 4 %âãÏÓ 1715 0 obj > endobj xref 1715 21 0000000016 00000 n 0000014010 00000 n 0000014079 00000 n 0000014217 00000 n 0000014306 00000 n 0000014459 00000 n 0000014612 00000 n 0000015048 00000 n 0000028413 00000 n 0000028817 00000 n 0000029193 00000 n 0000029556 00000 n 0000029708 00000 n 0000030380 00000 n 0000048618 00000 n 0000049244 00000 n 0000049723 00000 n 0000050089 00000 n 1. Choose Class: There is a list of classes, but you have to choose class 11 in order to download the Probability Class 11 Notes PDF. Weekly Demand of a Slow-Moving Product. Let P and Q be any two events, then the following formulas can be derived. ( 20/50 = 0. De- Probability Notes for Class 11. S. Denote by #Adenote the number of point in A. 1 They are not intended to stand alone. docx), PDF File (. Probability is the measure of the likelihood of an event to occur. Moment Generating unctionsF 93 Chapter 14. The sum of the probabilities of all the elementary events of an experiment is 1. probability in an earlier version of this book. Bookmark the permalink . On this page you will get important Probability PDF for SBI Clerk Mains, IBPS Clerk Mains, CET, LIC AAO, SBI PO Pre and IBPS PO Pre and IBPS RRB Officer Scale 1 It is convenient to introduce the probability function, also referred to as probability distribution, given by P(X x) f(x) (2) For x x k, this reduces to (1) while for other values of x, f(x) 0. #C = #S + #T. When you flip a coin, the probability of getting a head is 50% - the probability of getting a tail is also 50%. In chapter 16 we will be going through the probability and outcomes concepts in Probability Class 11 notes. #C = mk. Find the probability that the vehicle was a van GIVEN THAT it was used. Noel Vaillant’s www. 4 (Probability measure). P(A0) = 1 P(A) Example 2. The text of the notes is quite polished and complete, but the prob-lems are less so. fis the frequency, xis the data item, and nis the number of data items in the set of data. Therefore, the probability of event A is: P (A) = n (A)/n (S) Where n (A) = Number of elements on the set A. 9) = 1: (1. Class 10 Maths Chapter 15 Probability MCQs. These notes focus on a narrow range of topics and give more weight to the probability than the statistics. 20 in a row is approximately 0:0001%, about 1 chance in a million. And the probability of picking a red ball in the above example is 3/10, as we already noted, because there are three desired outcomes (picking any of the three red balls) and No. MC Gillem, Oxford, 3rd Edition, 1999. This book is an introductory textbook in undergraduate probability. At the risk of seeming overly-dramatic, I like to think of it as the set of all possible timelines of all possible These course notes accompany Feller, An Introduction to Probability Theory and Its Applications, Wiley, 1950. ) (PDF) 39. Bivariate Expectations, conditional mean and variance, covariance, Karl Pearson Correlation coefficient, independence of random variables based on expectation. (i) Trial: A single performance of a random experiment is known as a trial. This is a crucial chapter for JEE Main Exams, and you can expect at least two questions from it. we refer to it as a probability space and often write this as (Ω,F,P). Cooper, Clave D. The Probability Density Function (PDF) is the probability function which is represented for the density of a continuous random variable lying between a certain range of values. pack. 4 days ago · Vedantu's Probability Class 10 Notes for CBSE Maths Chapter 15 offer a comprehensive and well-structured resource for students seeking to master this fundamental concept. 4 days ago · Revision Notes for CBSE Class 12 Maths Chapter 12 (Linear Programming) - Free PDF Download. pdf: 244. Then a probability distribution or probability density function (pdf) of X is a function f (x) such that for any two numbers a and b with a ≤ b, we have The probability that X is in the interval [a, b] can be calculated by integrating the pdf of the r. The second axiom states that (a) the probability of an event A S must not exceed one, and (b) the probability that at least one elementary event s in the sample space S occurs must equal one. Stanford University Probability Introduction Notes IIT JEE Probability Notes – Formulas. The former approach is developed using the foundation of set theory, and a quick review of the theory is in order. The. For any event E, P (E) + P ( E¯) = 1, where E¯ stands for ‘not E’. 6 Bernoulli random variables 3. They are called indicators. Mar 22, 2022 · Download PDF. Jonathan Kuhn, Ph. • It should be 25 heads. Each of the three colors has positive probability. Calculate the probability of an event by making a sum of 1. 3. Students can download the Probability revision notes PDF from Vedantu or refer to the notes online. If these permutations are generated randomly with equal probability then what is the probability the word starts with the letter ”a ” ? SOLUTION : 2 6 = 1 3. e. 87 12. Bayes' theorem. Equally Likely Events: The given events are said to be equally likely if none of […] Set books The notes cover only material in the Probability I course. v. The random variables for which S= f0;1gare especially useful. Class 11 Math chapter 16 notes are regarding Probability. The interrupted game of chance (Fermat). Mar 10, 2024 · Class 10 Notes for Maths – All Chapters. It measures the uncertainty of happening something. Multiplication Principle. 56) • Please compare their results with the theoretical probability. tiation and integration is a necessary prerequisite to this course. Limit Laws 97 14. Conditional Probability Suppose that green ball was observed in the second draw. This text is designed for an introductory probability course taken by sophomores,juniors, and seniors in mathematics, the physical and social sciences, engineering,and computer science. Day One- Introduction to Probability Use informal measures of probability. Probability and Statistics bca 3rd sem 3rd semester third sem bca third semester Probability and Statistics question paper model question paper yearly question paper mcq bca 3rd sem bca 3rd semester These are lecture notes intended for teaching MATH 5010: Introduction to Probability at the University of Utah. Variance and covariance. Check out how many questions were asked from the Statistics and Probability chapter in CBSE Class 9 Maths Probability Notes:-Download PDF Here Introduction to Probability Probability. Readers familiar with the basics of set theory such Solution. • Probability and Statistics for Engineering and the Sciences by Jay L. 8 Geometric random variables 3. Also, the marking scheme and answer key for each paper is available. The standard normal distribution is used to create a database or To do this, click on NCERT Notes within the NCERT Books and Solutions menu. Clue #1. Step 3: To find probability, divide n (A) by n (S). Further examples appear in later lessons. f(x) 0 2. This ratio can be expressed as a common fraction, a decimal fraction or a percentage. 6. 8) = total total (1. What is the probability th. Identify and distinguish experiment, trials, outcomes, and events. You pick one urn at random and then select a ball from the urn. (lii) Event: A subset of the sample space of a random experiment is called an event. Many of these comments were very thought-provoking; in addition, they provided a student’s perspective on the book. 2 Probability mass functions (PMF) 3. Now, Select Maths and Access Probability Notes for Free: At this moment, you are two steps away from downloading your revision notes. What is the conditional probability that the first ball was also green? It is given by P(G1jG2) = P(G1G2) P(G2) = 2 30 10 30 0:2 Exercise 1. Most of the major changes in this version of the book have their genesis in these notes. It defines events, establishes probability for equally likely outcomes (the ‘equiprobable model’) and gives a brief example. Leave a comment Cancel reply To do this, click on NCERT Notes within the NCERT Books and Solutions menu. Probability Spaces, Random Variables, and Expectation Probability spaces. ) (PDF) 40. where. where the sum in 2 is taken over all possible values of x. If a collection C of objects can be divided into two collections S and T which have no objects in common, then. This is a 8 day unit on Probability. If Ω is a finite set with npoints and we define P(A) = 1 n #A then we get a probability model is a description of some chance process that consists of two parts: a sample space S and. • Lisa had a head 20 times. May 21, 2019 · CBSE Class 11 Maths Notes Chapter 16 Probability. use our Materials to score good marks in the examination. The number of distinct combinations of 3 professors is. The textbook for this subject is Bertsekas, Dimitri, and John Tsitsiklis. Event P or Q: The set P ∪ Q. Example 1: Find the probability of getting a number less than 5 when a dice is rolled by using the probability formula. To calculate the probability of an event, we simply need to find out the total number of possible outcomes of an experiment and the number of outcomes which correspond to the given event. Apr 12, 2024 · Class 12 Probability Maths Sample Paper or the board exam 2024-25 has been released by the Central Board of Secondary Education (CBSE). To each element x of the sample space, we assign a probability, which will be a non-negative number. These are written by Cliff Taubes (who developed the course), but re-formatted and slightly revised for Spring 2010. In tossing a coin, getting either head or tail is an event. 4) • Tom had a head 26 times. The conditional probability of an event with respect to another event is the probability of occurrence of the event comes of the roll of a die, or ips of a coin. R. These notes are intended to accompany the textbook of the course. Regulation. Define µ(A) = #A. The probability that a drawing pin will land ‘point up’ is 0:62. 3Properties of a PMF 3. So probability can be used, and it can be (and quite frequently is) abused Example 1. Expectation of Sums of R. The Selfstudys website provides the ICSE Class 10 Probability notes PDF which is in the online mode and accordingly students can access the study material from their comfort zone. Expectations 87 12. Probability can be expressed in terms such as impossible, unlikely, likely, or certain, as a number between 0 and 1, or as a percent between 0% and 100% as illustrated on the number line. e. Follow the links given on this website to download the notes. number of possible outcomes. The probability that a fair coin will land heads is 1=2. List of all possible [ xi, p(xi) ] pairs. That is less a value judgment and more a reflection of my limited knowledge. An event having only one outcome is called an elementary event. Example 2. 5) our intuition, and they make probability an extremely useful tool in all the sciences. etween 0 and 1, note by p(x). Definition 8. Review for Final Exam (cont. This Class 11 Maths chapter 16 notes contains the following topics: sample space, outcomes, events, types of events like simple, exclusive, exhaustive, exclusive, and 1. Jan 23, 2024 · CBSE Class 9 Maths Notes Chapter 15 Probability. : The provided notes have been created by the experts. probability. Joint PDF of functions 84 Chapter 12. The phrases 'probably', 'doubt’, ‘most probably', 'chances' and so on, all have an element of ambiguity in them. Sometimes, we know the conditional probability of E1 given E2, but we are interested in the conditional probability of E2 given E1. These notes completely follow the latest syllabus of GATE CS 2021. Each day has a Power Point that includes a warm -up with answers, notes and a closure of the lesson. One way to find the mean is to list all the values in the frequency table, add up these values, and then divide the answer by the number of values. Probability is a measure of one's belief in the occurrence of a future event. Let Ω be a countable set and let F = collection of all subsets of Ω. Probability is a very important topic from any examination perspective. i. If you get 5 heads in a row, the probability of getting a tail Jun 7, 2023 · MA3391 PS Notes : We are providing the MA3391 Probability and Statistics Lecture Notes, and Hand Written Notes Collections PDF below for your examination success. Feb 21, 2024 · So if you want to be master of Probability Questions you must download this PDF with 50 problems as much as you can make your maths basic strong you can solve Bank Exams Questions Fast. Probability that something will not happen. Here the total probability is just two terms: P(A) = P(AjB)P(B) + P(AjBc)P(Bc) In-Class Problem: You have two urns, one with 4 black balls and 3 white balls, the other with 2 black balls and 2 white balls. a probability for each outcome. For discrete random variables, individual points can have P(X = x) 0. In general, f(x) is a probability function if 1. Here are two basic counting principles to keep in mind: Addition Principle. You will be able to get this in PDF file format for Free of cost. 2 Contrast Experimental and theoretical probability Three students tossed a coin 50 times individually. 52) • Al had a head 28 times. Probability Density Function explains the normal distribution and how mean and deviation exists. If you are studying Probability, then chances are that you will study the other chapters in class 10 Maths. of numbers greater than 2 = 6. Nov 26, 2015 · PDF | This documents contain some basic concepts of probability theory Lecture notes for preliminary level of students | Find, read and cite all the research you need on ResearchGate If the probability that an event occurs is p, then the probability that the event does not occur is q= (1 p). 3 Cumulative distribution functions 3. CIE A Level Maths: Probability & Statistics 1Revision Notes. And they make the study of probability much more fun, too! A note of caution is in order, though: mathematical models are only as good as the assumptions they are based on. If Mr. The Central Limit Theorem 97 2 SET THEORY AND LOGIC & ELEMENTRY PROBABILITY THEORY Definition 1. Brown chooses a student to answer a question in his classroom, what is the probabil. Otherwise, the probability of +1is positive. Students must Download the complete Class 12 Maths Sample Papers in PDF for the final examination’s excellent score. net tutorials; Dmitry Panchenko’s notes for an earlier rendition of 18. 1 on the second die. The probability that a selection of 6 numbers wins the National Lottery Lotto jackpot is 1 in 49 6 =13,983,816, or 7:15112 10 8. 11 kB: Adobe PDF: View/Open: Show full item record ICSE Class 10 Probability Notes PDF. What this means intuitively is that when we perform our process, exactly Overview of Probability. All random variables are assumed to take values in R. Brown’s. Some familiarity with both di ere. A possible result of a random experiment is called its outcome. If a collection C can be divided into m types and each type can be broken up into k subtypes, then. This document introduces key concepts in probability theory. 1: Joint pmf and joint pdf, marginal and conditional probability, independence of random variables, function of random variable. Flipped coin. The text-books listed below will be useful for other courses on probability and statistics. X. Woods, Pearson Education, 3rd Edition. 4 Expectation 3. The Statistics and Probability chapter covers the mathematical study of data analysis, probability theory, and statistical methods. by. Calculate the probability of an event by creating a ratio. Mutually exclusive (nothing in common) Collectively exhaustive (nothing left out) 0 p(xi) 1 P(xi) = 1. These notes are not only a reference but a lecture tool. Find the probability that a vehicle was used GIVEN THAT it was a van. For example: When a fair 6-sided die is rolled, the Sample Space is S = {1, 2, 3, ,4,5, 6}. Find the probability that a vehicle is NOT a truck. We require thatX p(x) = 1;x2Sso the total probabi. Download Course. (ii) Sample space: The set consisting of all possible outcomes of a random experiment is known as sample space. 52 P (B) = 4 since there are 4 aces in the. So a probability of 5 out of 8 can be written as 5 / 8 or as 0,625 or as 62,5%. While any series of 20 ips is a one in a million probability, we only tend to notice those that have a distinct pattern. pdf), Text File (. This is called the counting measure. 52. 4. Measure preserving transformations, Bernoulli shifts. , and a, b are integers with b a. Scott Sheffield. Final Exam (No Lecture) Instructor. De nition 1. The probability of getting any number on a 6-sided die is fixed at 1 6, regardless of whatever numbers you have already rolled. A probability measure is a function de ned over the sets in a we are sharing bca third semester complete notes model question paper yearly question paper MCQ question paper , lab reports, mini project, and related all thing of third semester all subject wise. 90 Chapter 13. V. • Warm Up in Power Point. Outcome. Spinning a blue is less likely than spinning a red. Another way is to work directly with the table using the formula = . Sample space: Collection of all possible outcomes of a random experiment is known as sample space Sample space is denoted by :. EXAMPLE : In general, if the word length is n and all characters are distinct then there are n! permutations of the word. Mark Kernighan contributed 40 pages of comments on the earlier edition. Data Presentation & Interpretation. n (S) = Total number of outcomes or the number of elements in the sample space S. 2Histogram 3. 3. 1 Everything you need to know about a PMF 3. What is the probability the ball is white? Feb 29, 2024 · Figure 1: Graph of pdf for \(X\), \(f(x)\) So, if we wish to calculate the probability that a person waits less than 30 seconds (or 0. ( Why ? ) If these permutations are generated randomly with Block-1 Basic Concepts in Probability: Files in This Item: File Description Size Format ; Unit-1. 2. This means that the endpoints of intervals ARE important for discrete random variables. 3 days ago · Chapter-wise Notes with PDF. his dart will land in a shaded square. Gaussian random variables, the multivariate normal distribution. If it happens to be equal to 1, you can safely conclude that X never takes the value +1. notes Lecture Notes. 2021. The notes are created by our subject matter experts according to the latest ICSE Class 10 syllabus; through this students can probability of rollinganevennumberonadieis1/2, becausetherearethreedesired outcomes (2, 4, and 6) and six total possible outcomes (the six numbers). In-formally, this is the collection of all possible outcomes or results of an experiment. 7) In Mr. Events can’t be predicted with certainty but can be expressed as to how likely it can occur using the idea of probability. The formula to find the combined probability of events obtained with replacement, and we simply find the product of the probabilities of the event directly: P (A ∩ B) = P (A) P (B) 2. partment, Purdue University Northwestc by Jonathan KuhniiPrefaceThe point of. Here are some examples that well describe the process of finding probability. The probability that a large earthquake will occur on the San Andreas Fault in Probability Distributions for Continuous Variables Definition Let X be a continuous r. It has a mission: to spell out the motivation, intuition, and implication of the probabilistic tools we use in science and engineering. Through their free PDF download, Vedantu provides a valuable aid to learners, simplifying complex probability principles with clarity and precision. Learn about equally likely outcomes, conditional probability, independence, and more. Two approaches of the concept of probability will be introduced later in the book: The classical probability and the experimental probability. classroom, 40% of the students are boys. 7 Binomial random variables 3. Additionally, the probability of the whole sample space should equal one, as it contains all outcomes P() = outcomes in total (1. Varadhan’s lecture notes; Amir Dembo’s lecture notes (PDF) Rick Durrett’s book at CiteSeer (PDF) or at Amazon and here is a recently updated version (PDF) from Durrett’s web page.  w2W Pr(w)=1. Probability and Random Processes with Application to Signal Processing - Henry Stark and John W. Choose Class: There is a list of classes, but you have to choose Class 10 in order to download the Probability Class 10 Notes PDF. (25/50 = 0. P (A) = 13 since there are 13 hearts in the pack. g. menu. txt) or read online for free. Further, suppose we know that if a person has lung cancer Lecture Notes #4 April 13, 2020 Conditional Probability Based on a chapter by Chris Piech 1 Conditional Probability In English, a conditional probability answers the question: “What is the chance of an event E happening, given that I have already observed some other event F?” Conditional probability onnie is playing darts with his friends. Clue #2. 5 days ago · PDF Summary - Class 11 Maths Probability (Chapter 16) Probability is a numerical measure of the uncertainty of diverse phenomena. Using the nPr notation, from a set of 3 objects we are choosing 3. Sample space S= f1;2;3;4;5;6gand event B= f1;2;4;6g. Note that the probability axioms should be interpreted as follows: The rst axiom states that the probability of an event A S must be non-negative. 1. probability of the complement of an event = 1 probability of the event. d. %PDF-1. The last roll of the game in backgammon (splitting the stakes at Monte Carlo). 3 P 3! 3 = = = = 3! 3! 3! • In our list of 210 sets of 3 professors, with order mattering, each set of three profs is counted 3! = 6 times. doc / . Find also (a): the probability that exactly one ball selected is green. From over half a decade of teaching the Aug 23, 2021 · We can work out the probability using the formula: Probability = number of favorable outcomes. I TheSample Space Some sources and uses of randomness, and philosophical conundrums. Describe the probability of an event using a number line. I, the in-structor, use the notes while also provide copies to the students. (b): the probability that at least one ball Discrete Probability Distribution. The course is attended by a large number of undergraduate and graduate The probability of ipping 1 head is 50%. CBSE Class 12 Maths Notes Chapter 13 Probability plays a crucial role in scoring good marks in maths. 1. eSaral helps the students by providing you an easy way to understand concepts and attractive study material for IIT JEE which includes the video lectures & Study Material designed by expert IITian Faculties of KOTA. Find the probability that the vehicle was a car AND was leased. The probability for a fair die would include the probabilities of these outcomes, which are all the same. f. S. this course is to introduce the mathematical theory of probability. Xi = value of random variable P(xi) = probability associated with value. A finite discrete probability space (or finite discrete sample space) is a finite set W of outcomes or elementary events w 2 W, together with a function Pr: W ! R, called probability measure (or probability distribution) satisfying the following properties: 0 Pr(w) 1 for all w 2W. Basic Definitions of Probability is the first in a series on lessons developing the foundations of probability theory. A free textbook for a course on elementary probability theory and applications, with examples, problems, and solutions. The strong law of large numbers, proof for independent random variables with bounded fourth moments. Probability Methods of Signal and System Analysis- George R. The probability of 10 in a row is about 0:1%. is ty oq bg rl mo yr sc rr rl