Math : Probabilities


 Erik Robertson
 2 years ago
 Views:
Transcription
1 Probability EPProgram  Strisuksa School  Roiet Math : Probabilities Dr.Wattana Toutip  Department of Mathematics Khon Kaen University 200 :Wattana Toutip Probability of successive events A Tree Diagram shows the probability of successive events 20.. Examples. A bag contains 5 red and 6 black marbles. Two are drawn without replacement. What is the probability that : (a) Both are red (b) They are different colors. Solution Construct the diagram as shown. The top branch corresponds to both marbles being red. Multiply the probabilities along this branch. Fig 20. (a) The Probability that both are red is The two middle branches correspond to the marbles having different colors (b) P(different colors) Four fair dice are rolled. Find the probabilities of (a) Four 6s (b) at least one 6. Solution (a) The Probability of four 6s is 6 The Probability of four 6s is (b) There will be at least one 6 unless none of the dice show a 6. This has probability The probability of at least one 6 is
2 20..2 Exercises. Two fair dice are rolled. Find the probabilities that: (a) Both are sixes (b) The total is 2 (c) The total is 7 (d) The first is greater than the second (e) A 'double' is thrown (f) At least one of the dice is a six. 2. Two card are drawn without replacement from a wellshuffled pack. Find the probabilities that : (a) The first is a heart (b) Both are heart (c) The first is a heart and the second is a spade (d) The first is a King and the second is a Queen. 3. A bag contains 5 blue are 4 green counters. Two are drawn without replacement. Find the probabilities that : (a) Both are blue (b) They are the same colour (c) There is at least one blue (d) The second is a green. 4. A sweet box contains 5 toffees, 6 liquorices and 8 chocolates. Two are drawn out. Find the probabilities that : (a) The first is a toffee and the second is a chocolate (b) At least one is a liquorice (c) Neither is a toffee. 5. Two fair dice are rolled. The score is the larger of the numbers showing. Find the probabilities that: (a) The score is (b) The score is 6 6. Two children A and B each pick at random a single digit from to 9. Find the probabilities that : (a) They pick the same number (b) A's number is larger than B's 7. To start a certain board game a die is rolled until a six is obtained. Find the probabilities that : (a) A player starts on his first roll (b) He starts on his second roll (c) He starts on his third roll (d) He has not started by his fourth roll 8. To start at darts a' double' must first be thrown. Albert has probability of throwing a 0 double, and Beatrice has probability 8. Albert throws first. Find the probabilities that : (a)both start on their first throw (b) Beatrice starts on her second throw but Albert has not started by then. 9. A fair coin is spun five times. Find the probabilities of (a) five Heads (b) at least one Head. 0. A roulette wheel has the number to. A gambler bets that a number divisible by 3 will turn up. The bet is repeated four times. Find the probabilities that the gambler (a) Wins all his bets (b) wins at least one bets.
3 . Five people take the driving test. Each has probability of passing. Find the probabilities that : (a) They all pass (b) at least one fails Exclusive and independent events. Conditional probability If two events cannot happen together, then they are exclusive. If events are exclusive then the probability that one or the other occurs is the sum of their probabilities. P A or B P A P B, provided that A and B are exclusive. Two events are independent if the truth of the one of them does not alter the probability of the other. If events A and B are independent then the probability of them both occurring is the product of their probabilities. PA & B PA PB The conditional probability of 6! 720 A given B is the probability of A, once it is known that B is true. The conditional probability of A given B is written PA B. It is obtained from the formula : PA B PB If A and B are independent then PA B PA P A & B The symbols and are sometimes used for 'and' and 'or' respectively Example. Two cards are drawn without replacement from a pack. Events A, B, C are as follows : A : the first is a heart. B : the second is a heart. C : the card is a king. Which pairs of these events are independent? Solution If A is true, then there is one fewer heart in the pack. The probability of B is 2. Hence 5 4 A and B are not independent. If A is true then C is neither more likely nor less likely than before. Its probability is still 3. Hence A and C are independent. Similarly the truth of B does not alter probability of C. The pairs A and C, B and C are independent 2. A women travels to work by bus, car or on foot with probabilities,, respectively. For each type of transport her probabilities of being late are,, arrives late one morning, find the probability that she come by bus. Solution respectively. If she
4 Here let L be the event that she is late, and B the event that she came by bus. Use the formula for conditional probability: P( B & L) PB L = PL ( ) P( B L) The probability that she came by bus is Exercises. Two cards are drawn without replacement from a pack. Events A, B,C,D are as follows : A : the first is an ace. B : the second is an 8. C : the first is red. D: the second is an spade Which pairs of events are independent? Which are exclusive? 2.Two dice are rolled. Events A, B, C, D are as follows: A : the first is an 5 B : the total is 8 C : the total is 7 D: the dice show the same number Which pairs of events are independent? Which are exclusive? 3. Two dice are rolled. Events A, B, C, D are as follows : A : the total is 7 B : the second die is a 2 C : both dice are less than 5 D: at least one die is a 6 Which pairs of events are independent? Which are exclusive? 4. Three fair coins are spun. Events A, B, C, D are as follows : A : the first coin is a head B : all the coins are heads C : there is at least one tail D: the first and last coins show the same Which pairs of events are independent? Which are exclusive? 5. With A, B, C, D as defined is Question, Find the following: (a) PB A (b) PD C 6. With A, B, C, D as defined is Question 2, Find the following: (a) PA B (b) PB D 7. With A, B, C, D as defined is Question 3, Find the following: (a) PA B (b) PC D 8. With A, B, C, D as defined is Question 4, Find the following: (a) PB A (b) PC A 9. A box contains 5 red and 6 blue marbles. Two are drawn without replacement. If the second is red find the probability that first was blue. 0. In his drawer a man has 7 left shoes and 0 right shoes. He picks two out at random. Find the probability that: (a) He has one left shoe and one right shoe (b) He has one left shoe and one right, given that the first was a left (c) The second is a left, given that he has one has one left and one right.
5 . A man travels to work by bus, car and motorcycle with probabilities 0.4, 0.5, 0. respectively. With each type of transport his chances of an accident are,, respectively. Find the probabilities that: (a) He goes by car and has an accident. (b) He does not have an accident. (c) He went by motorcycle, given that he had an accident. 2. % of the population has a certain disease. There is a test for the disease, which gives a positive response for 9 of the people with the disease, and for of the people without the 0 50 disease. A person is selected at random and tested. (a) What is the probability that the test gives a negative response? (b) If the test is positive, what is the probability that the person has the disease? (c) If the test is negative, what is the probability that the person does not have the disease? 3. An island contains two tribes; 2 3 of the population are Wache, who tell the truth with probability 0.7, and 3 are Oya, who tell the truth with probability 0.8. I meet a tribesman who tell me that he is a Wache. What is the probability that he is telling the truth? 4. In the certain town 3 4 of the voters are over 25, and they vote the Freedom Party with probability. Voters under 25 support the Freedom Party with probability 3. If a support of the Freedom Party is picked at random, what is the probability that he or she is under 25? 5. Events A and B are such that PA 0.4, PB 0.3, and Show that A and B are neither exclusive nor independent. Find PA B. 6. Events A and B are such that PA 0.3, PB 0.2, and PA or B 0.4 and B are not exclusive. Find PA & B and PA B. P A & B Show that A 7. Events A and B are such that PA 0.4, PB 0.3, and PA B 0.5. Find PA & B and PA or B Find PB A 8. Events A and B are such that PA 0.3, PB 0.5. Find PA or B and PA & B in the following cases: (a) A and B are exclusive (b) A and B are independent 20.3 Examination questions. A 2p coin a 0p are throw on a table. Event A is 'A head occurs on the 2p coin'. Event B is 'A head occurs on the 0p coin'. Event C is 'Two head or two tails obtained'. State, giving reasons, which of the following statements is (are) true and which is (are) false. (a) A and B are independent events (b) B and C are independent events. (c) A and C are mutually exclusive events
6 (d) A and BC are independent events. (e) PA B C = PA.P B.P C 2. Six balls colored yellow, green, brown, blue, pink, and black have values 2, 3, 4, 5, 6, 7 respectively. They are independent in size and placed in a box. Two balls are selected together from the box at random and the total number of point recorded (i) Find the probability that the total score is (a) 7, (b)9, (c)0, (d) greater 9, (e) odd. (ii) A game between two players, X and Y, starts with the six balls in a box. Each player in turn selects at random two balls, notes the score and then returns the balls to the box. The game is over when one of the players reaches a total score of 25 or more. (a) If X starts, calculate the probability that X wins on his second turn; (b) If Y starts, calculate the probability that Y wins on his second turn. [O ADD] 3. (a) The two electronic systems C,C2of a communications satellite operate independently and have probabilities of 0.and 0.05respectively of failing. Find the probability that (i) neither circuit fails (ii) at least one circuit fails, (iii) exactly one circuit fails (b) In a certain boxing competition all fights are either won or lost; draws are not permitted. If a boxer wins a fight then the probability that he wins his next fight is 3 ; if he loses 4 a fight the probability of him losing the next three fights is 2 3. Assuming that he won his last fight, use a tree diagram, or otherwise, to calculate the probability that of his three fights (i) he wins exactly two fights (ii) he wins at most two fights. State the most likely and least likely sequence of results for these three fights. 4. (i) The events A and B are such that PA 0.4, PB 0.45, P A B Show that the events A and B are neither mutually exclusive nor independent. (ii) A bag contains 2 red balls, 8 blue balls and 4 white balls. Three balls are taken from the bag at random and without replacement. Find the probability that all three balls are of the same colors. Find also the probability that all three balls are of different colors. 5. A box contains 25 apples, of which 20 are red and 5 are green. Of the red apples, 3 contains maggots and of the green apples, contains maggots. Two apples are chosen at random from the box. Find, in any order, (i) the probability that both apples contain maggots, (ii) the probability that both apples are red and at least one contains maggots, (iii) the probability that at least one apple contains maggots, given that both apples are red,
7 (iv) the probability that both apples are red given that at least one apple is red. 6. (a) Two digits X and Y are taken from a table of random sampling numbers. Event R is that X Y and events S is that X and Y are both less than 2. Write down (i) PR (ii) PRS (iii) PRS (iv) PR S (b) Conveyor belting for use in mines is tested for both strength and safety (the safety test is based on the amount of heat generated if the belt snaps). A testing station receives belting from three different suppliers : 30% of its tests are carried out on samples of belting from supplier A, 50% from B ; 20 % from C. From past experience the probability of failing the strength test is 0.02 for a sample from A, 0.2 from B and 0.04 from C. (i) What is the probability that a particular strength test will result in a failure? (ii) If a strength test result in a failure, What is the probability that the belting came from supplier A? (iii)what is the probability of a sample failing the safety tests given the following further information: supplier A  the probability of failing the safety tests is 0.05 and is independent of the probability of failing the strength test; supplier B % the probability of samples fail both strength and safety test supplier C exactly half the samples which fail the strength test also fail the safety test Common errors. Single probability If there are n outcomes to an experiment, then each has probability only if they are equally likely When two dice are rolled, there are possible for total score. But a total of 2 is less likely than total of 7,so neither has probability 2. Addition of probability The probability of 'A or 'B is only the sum of the probabilities if the events concerned are exclusive. In general: PA or B PA PB 3. Multiplication of probabilities The probability of A & B is only the product of the probabilities if the events concerned are independent. 4. Conditional probability independent. (a) Do not forgot to divide by PB when working out PA B (b) In the formula P A B do not assume that PA & Bis PA PB P( A& B) PB ( ) This is only true if A and B are
8 (c) Conditional probability is concerned with belief, not with cause and effect. If PA P A B. then it does not follow that B has caused A or prevented A. It may even be that B happened after A did. Solution (to exercise) (a) (e) 6 2. (a) 4 3. (a) (a) (a) (f) (b) 7 (b) 3 57 (b) (b) (b) 4 9 (c) (c) 9 7 (c) 6 (c) 5 6 (d) 5 2 (d) (d) (a) 9 (b) (a) 6 8. (a) (a) 32 (b) 5 (b) (b) 3 32 (b) 65 8 (b) (a) 8. (a) A& C, A& D, B & D indep. None exc. 2. A& C, A& D, B & D indep. B & C, C & D excl. 3. A& B indep. C& D exclusive. 4. A& Dindep. B& C excl. 5. (a) 4 (b) (c) (d) (a) 5 7. (a) 5 (b) 6 (b) 0
9 8. (a) 4 (b) (a) (b) 5 8 (c) 2. (a) (b) 5 8 (c) 2 2. (a) (b) (c) , ,0.55, (a) 0.8,0 (b) 0.65,0.5 =========================================================== References: Solomon, R.C. (997), A Level: Mathematics (4 th Edition), Great Britain, Hillman Printers(Frome) Ltd. More: (in Thai)
Topic : ADDITION OF PROBABILITIES (MUTUALLY EXCLUSIVE EVENTS) TIME : 4 X 45 minutes
Worksheet 6 th Topic : ADDITION OF PROBABILITIES (MUTUALLY EXCLUSIVE EVENTS) TIME : 4 X 45 minutes STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of
More informationModule 4 Project Maths Development Team Draft (Version 2)
5 Week Modular Course in Statistics & Probability Strand 1 Module 4 Set Theory and Probability It is often said that the three basic rules of probability are: 1. Draw a picture 2. Draw a picture 3. Draw
More informationMEP Practice Book SA5
5 Probability 5.1 Probabilities MEP Practice Book SA5 1. Describe the probability of the following events happening, using the terms Certain Very likely Possible Very unlikely Impossible (d) (e) (f) (g)
More informationChapterwise questions. Probability. 1. Two coins are tossed simultaneously. Find the probability of getting exactly one tail.
Probability 1. Two coins are tossed simultaneously. Find the probability of getting exactly one tail. 2. 26 cards marked with English letters A to Z (one letter on each card) are shuffled well. If one
More informationRevision Topic 17: Probability Estimating probabilities: Relative frequency
Revision Topic 17: Probability Estimating probabilities: Relative frequency Probabilities can be estimated from experiments. The relative frequency is found using the formula: number of times event occurs.
More informationMEP Practice Book ES5. 1. A coin is tossed, and a die is thrown. List all the possible outcomes.
5 Probability MEP Practice Book ES5 5. Outcome of Two Events 1. A coin is tossed, and a die is thrown. List all the possible outcomes. 2. A die is thrown twice. Copy the diagram below which shows all the
More informationChapter 16. Probability. For important terms and definitions refer NCERT text book. (6) NCERT text book page 386 question no.
Chapter 16 Probability For important terms and definitions refer NCERT text book. Type I Concept : sample space (1)NCERT text book page 386 question no. 1 (*) (2) NCERT text book page 386 question no.
More informationFunctional Skills Mathematics
Functional Skills Mathematics Level Learning Resource Probability D/L. Contents Independent Events D/L. Page  Combined Events D/L. Page  9 West Nottinghamshire College D/L. Information Independent Events
More informationQ1) 6 boys and 6 girls are seated in a row. What is the probability that all the 6 gurls are together.
Required Probability = where Q1) 6 boys and 6 girls are seated in a row. What is the probability that all the 6 gurls are together. Solution: As girls are always together so they are considered as a group.
More informationHARDER PROBABILITY. Two events are said to be mutually exclusive if the occurrence of one excludes the occurrence of the other.
HARDER PROBABILITY MUTUALLY EXCLUSIVE EVENTS AND THE ADDITION LAW OF PROBABILITY Two events are said to be mutually exclusive if the occurrence of one excludes the occurrence of the other. Example Throwing
More informationWorksheets for GCSE Mathematics. Probability. mrmathematics.com Maths Resources for Teachers. Handling Data
Worksheets for GCSE Mathematics Probability mrmathematics.com Maths Resources for Teachers Handling Data Probability Worksheets Contents Differentiated Independent Learning Worksheets Probability Scales
More informationPROBABILITY.0 Concept Map Contents Page. Probability Of An Event. Probability Of Two Events. 4. Probability of Mutually Exclusive Events.4 Probability
PROGRAM DIDIK CEMERLANG AKADEMIK SPM ADDITIONAL MATHEMATICS FORM MODULE PROBABILITY PROBABILITY.0 Concept Map Contents Page. Probability Of An Event. Probability Of Two Events. 4. Probability of Mutually
More informationClass XII Chapter 13 Probability Maths. Exercise 13.1
Exercise 13.1 Question 1: Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E F) = 0.2, find P (E F) and P(F E). It is given that P(E) = 0.6, P(F) = 0.3, and P(E F) = 0.2 Question 2:
More informationPROBABILITY Case of cards
WORKSHEET NO1 PROBABILITY Case of cards WORKSHEET NO2 Case of two die Case of coins WORKSHEET NO3 1) Fill in the blanks: A. The probability of an impossible event is B. The probability of a sure
More informationPROBABILITY M.K. HOME TUITION. Mathematics Revision Guides. Level: GCSE Foundation Tier
Mathematics Revision Guides Probability Page 1 of 18 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier PROBABILITY Version: 2.1 Date: 08102015 Mathematics Revision Guides Probability
More informationThe Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.)
The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you $ that if you give me $, I ll give you $2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If
More informationBefore giving a formal definition of probability, we explain some terms related to probability.
probability 22 INTRODUCTION In our daytoday life, we come across statements such as: (i) It may rain today. (ii) Probably Rajesh will top his class. (iii) I doubt she will pass the test. (iv) It is unlikely
More information, x {1, 2, k}, where k > 0. (a) Write down P(X = 2). (1) (b) Show that k = 3. (4) Find E(X). (2) (Total 7 marks)
1. The probability distribution of a discrete random variable X is given by 2 x P(X = x) = 14, x {1, 2, k}, where k > 0. Write down P(X = 2). (1) Show that k = 3. Find E(X). (Total 7 marks) 2. In a game
More information5.6. Independent Events. INVESTIGATE the Math. Reflecting
5.6 Independent Events YOU WILL NEED calculator EXPLORE The Fortin family has two children. Cam determines the probability that the family has two girls. Rushanna determines the probability that the family
More informationLesson 3 Dependent and Independent Events
Lesson 3 Dependent and Independent Events When working with 2 separate events, we must first consider if the first event affects the second event. Situation 1 Situation 2 Drawing two cards from a deck
More informationSection A Calculating Probabilities & Listing Outcomes Grade F D
Name: Teacher Assessment Section A Calculating Probabilities & Listing Outcomes Grade F D 1. A fair ordinary sixsided dice is thrown once. The boxes show some of the possible outcomes. Draw a line from
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Study Guide for Test III (MATH 1630) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the number of subsets of the set. 1) {x x is an even
More informationSTANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving.
Worksheet 4 th Topic : PROBABILITY TIME : 4 X 45 minutes STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving. BASIC COMPETENCY:
More information1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 100 calculators is tested.
1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 0 calculators is tested. Write down the expected number of faulty calculators in the sample. Find
More informationContents 2.1 Basic Concepts of Probability Methods of Assigning Probabilities Principle of Counting  Permutation and Combination 39
CHAPTER 2 PROBABILITY Contents 2.1 Basic Concepts of Probability 38 2.2 Probability of an Event 39 2.3 Methods of Assigning Probabilities 39 2.4 Principle of Counting  Permutation and Combination 39 2.5
More informationName. Is the game fair or not? Prove your answer with math. If the game is fair, play it 36 times and record the results.
Homework 5.1C You must complete table. Use math to decide if the game is fair or not. If Period the game is not fair, change the point system to make it fair. Game 1 Circle one: Fair or Not 2 six sided
More informationPROBABILITY. 1. Introduction. Candidates should able to:
PROBABILITY Candidates should able to: evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g for the total score when two fair dice are thrown), or by calculation
More informationExam III Review Problems
c Kathryn Bollinger and Benjamin Aurispa, November 10, 2011 1 Exam III Review Problems Fall 2011 Note: Not every topic is covered in this review. Please also take a look at the previous WeekinReviews
More informationExam 2 Review F09 O Brien. Finite Mathematics Exam 2 Review
Finite Mathematics Exam Review Approximately 5 0% of the questions on Exam will come from Chapters, 4, and 5. The remaining 70 75% will come from Chapter 7. To help you prepare for the first part of the
More informationMathematics 'A' level Module MS1: Statistics 1. Probability. The aims of this lesson are to enable you to. calculate and understand probability
Mathematics 'A' level Module MS1: Statistics 1 Lesson Three Aims The aims of this lesson are to enable you to calculate and understand probability apply the laws of probability in a variety of situations
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
6. Practice Problems Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the probability. ) A bag contains red marbles, blue marbles, and 8
More informationProbability  Grade 10 *
OpenStaxCNX module: m32623 1 Probability  Grade 10 * Rory Adams Free High School Science Texts Project Sarah Blyth Heather Williams This work is produced by OpenStaxCNX and licensed under the Creative
More informationConditional Probability Worksheet
Conditional Probability Worksheet P( A and B) P(A B) = P( B) Exercises 36, compute the conditional probabilities P( AB) and P( B A ) 3. P A = 0.7, P B = 0.4, P A B = 0.25 4. P A = 0.45, P B = 0.8, P A
More informationNotes #45 Probability as a Fraction, Decimal, and Percent. As a result of what I learn today, I will be able to
Notes #45 Probability as a Fraction, Decimal, and Percent As a result of what I learn today, I will be able to Probabilities can be written in three ways:,, and. Probability is a of how an event is to.
More informationIndependent Events B R Y
. Independent Events Lesson Objectives Understand independent events. Use the multiplication rule and the addition rule of probability to solve problems with independent events. Vocabulary independent
More informationHere are two situations involving chance:
Obstacle Courses 1. Introduction. Here are two situations involving chance: (i) Someone rolls a die three times. (People usually roll dice in pairs, so dice is more common than die, the singular form.)
More informationConditional Probability Worksheet
Conditional Probability Worksheet EXAMPLE 4. Drug Testing and Conditional Probability Suppose that a company claims it has a test that is 95% effective in determining whether an athlete is using a steroid.
More informationLC OL Probability. ARNMaths.weebly.com. As part of Leaving Certificate Ordinary Level Math you should be able to complete the following.
A Ryan LC OL Probability ARNMaths.weebly.com Learning Outcomes As part of Leaving Certificate Ordinary Level Math you should be able to complete the following. Counting List outcomes of an experiment Apply
More informationABC High School, Kathmandu, Nepal. Topic : Probability
BC High School, athmandu, Nepal Topic : Probability Grade 0 Teacher: Shyam Prasad charya. Objective of the Module: t the end of this lesson, students will be able to define and say formula of. define Mutually
More informationIndependent and Mutually Exclusive Events
Independent and Mutually Exclusive Events By: OpenStaxCollege Independent and mutually exclusive do not mean the same thing. Independent Events Two events are independent if the following are true: P(A
More informationKey Concepts. Theoretical Probability. Terminology. Lesson 111
Key Concepts Theoretical Probability Lesson  Objective Teach students the terminology used in probability theory, and how to make calculations pertaining to experiments where all outcomes are equally
More informationMath 1313 Section 6.2 Definition of Probability
Math 1313 Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability
More informationSTAT 311 (Spring 2016) Worksheet: W3W: Independence due: Mon. 2/1
Name: Group 1. For all groups. It is important that you understand the difference between independence and disjoint events. For each of the following situations, provide and example that is not in the
More informationWhen combined events A and B are independent:
A Resource for reestanding Mathematics Qualifications A or B Mutually exclusive means that A and B cannot both happen at the same time. Venn Diagram showing mutually exclusive events: Aces The events
More informationProbability QUESTIONS Principles of Math 12  Probability Practice Exam 1
Probability QUESTIONS Principles of Math  Probability Practice Exam www.math.com Principles of Math : Probability Practice Exam Use this sheet to record your answers:... 4... 4... 4.. 6. 4.. 6. 7..
More informationCLASSIFIED ALEVEL PROBABILITY S1 BY: MR. AFDZAL Page 1
5 At a zoo, rides are offered on elephants, camels and jungle tractors. Ravi has money for only one ride. To decide which ride to choose, he tosses a fair coin twice. If he gets 2 heads he will go on the
More informationIntermediate Math Circles November 1, 2017 Probability I
Intermediate Math Circles November 1, 2017 Probability I Probability is the study of uncertain events or outcomes. Games of chance that involve rolling dice or dealing cards are one obvious area of application.
More information4. Are events C and D independent? Verify your answer with a calculation.
Honors Math 2 More Conditional Probability Name: Date: 1. A standard deck of cards has 52 cards: 26 Red cards, 26 black cards 4 suits: Hearts (red), Diamonds (red), Clubs (black), Spades (black); 13 of
More information7.1 Experiments, Sample Spaces, and Events
7.1 Experiments, Sample Spaces, and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment
More informationTUESDAY, 8 NOVEMBER 2016 MORNING 1 hour 45 minutes
Surname Centre Number Candidate Number Other Names 0 GCSE NEW 3300U30 A63300U30 MATHEMATICS UNIT : NONCALCULATOR INTERMEDIATE TIER TUESDAY, 8 NOVEMBER 206 MORNING hour 45 minutes For s use ADDITIONAL
More information4.1 Sample Spaces and Events
4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an
More informationCSC/MATA67 Tutorial, Week 12
CSC/MATA67 Tutorial, Week 12 November 23, 2017 1 More counting problems A class consists of 15 students of whom 5 are prefects. Q: How many committees of 8 can be formed if each consists of a) exactly
More informationUnit 7 Central Tendency and Probability
Name: Block: 7.1 Central Tendency 7.2 Introduction to Probability 7.3 Independent Events 7.4 Dependent Events 7.1 Central Tendency A central tendency is a central or value in a data set. We will look at
More informationMTH 103 H Final Exam. 1. I study and I pass the course is an example of a. (a) conjunction (b) disjunction. (c) conditional (d) connective
MTH 103 H Final Exam Name: 1. I study and I pass the course is an example of a (a) conjunction (b) disjunction (c) conditional (d) connective 2. Which of the following is equivalent to (p q)? (a) p q (b)
More informationStat210 WorkSheet#2 Chapter#2
1. When rolling a die 5 times, the number of elements of the sample space equals.(ans.=7,776) 2. If an experiment consists of throwing a die and then drawing a letter at random from the English alphabet,
More informationStatistics 1040 Summer 2009 Exam III
Statistics 1040 Summer 2009 Exam III 1. For the following basic probability questions. Give the RULE used in the appropriate blank (BEFORE the question), for each of the following situations, using one
More informationInstructions: Choose the best answer and shade in the corresponding letter on the answer sheet provided. Be sure to include your name and student ID.
Math 3201 Unit 3 Probability Test 1 Unit Test Name: Part 1 Selected Response: Instructions: Choose the best answer and shade in the corresponding letter on the answer sheet provided. Be sure to include
More informationPage 1 of 22. Website: Mobile:
Exercise 15.1 Question 1: Complete the following statements: (i) Probability of an event E + Probability of the event not E =. (ii) The probability of an event that cannot happen is. Such as event is called.
More informationExercise Class XI Chapter 16 Probability Maths
Exercise 16.1 Question 1: Describe the sample space for the indicated experiment: A coin is tossed three times. A coin has two faces: head (H) and tail (T). When a coin is tossed three times, the total
More informationProbability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College
Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical
More informationMathacle. Name: Date:
Quiz Probability 1.) A telemarketer knows from past experience that when she makes a call, the probability that someone will answer the phone is 0.20. What is probability that the next two phone calls
More informationProbability. A Mathematical Model of Randomness
Probability A Mathematical Model of Randomness 1 Probability as Long Run Frequency In the eighteenth century, Compte De Buffon threw 2048 heads in 4040 coin tosses. Frequency = 2048 =.507 = 50.7% 4040
More informationSection Introduction to Sets
Section 1.1  Introduction to Sets Definition: A set is a welldefined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase
More informationProbability. Probabilty Impossibe Unlikely Equally Likely Likely Certain
PROBABILITY Probability The likelihood or chance of an event occurring If an event is IMPOSSIBLE its probability is ZERO If an event is CERTAIN its probability is ONE So all probabilities lie between 0
More informationPROBABILITY. Example 1 The probability of choosing a heart from a deck of cards is given by
Classical Definition of Probability PROBABILITY Probability is the measure of how likely an event is. An experiment is a situation involving chance or probability that leads to results called outcomes.
More informationKS3 Questions Probability. Level 3 to 5.
KS3 Questions Probability. Level 3 to 5. 1. A survey was carried out on the shoe size of 25 men. The results of the survey were as follows: 5 Complete the tally chart and frequency table for this data.
More informationMath June Review: Probability and Voting Procedures
Math  June Review: Probability and Voting Procedures A big box contains 7 chocolate doughnuts and honey doughnuts. A small box contains doughnuts: some are chocolate doughnuts, and the others are honey
More informationSection 7.3 and 7.4 Probability of Independent Events
Section 7.3 and 7.4 Probability of Independent Events Grade 7 Review Two or more events are independent when one event does not affect the outcome of the other event(s). For example, flipping a coin and
More informationRANDOM EXPERIMENTS AND EVENTS
Random Experiments and Events 18 RANDOM EXPERIMENTS AND EVENTS In daytoday life we see that before commencement of a cricket match two captains go for a toss. Tossing of a coin is an activity and getting
More informationTEST A CHAPTER 11, PROBABILITY
TEST A CHAPTER 11, PROBABILITY 1. Two fair dice are rolled. Find the probability that the sum turning up is 9, given that the first die turns up an even number. 2. Two fair dice are rolled. Find the probability
More informationContemporary Mathematics Math 1030 Sample Exam I Chapters Time Limit: 90 Minutes No Scratch Paper Calculator Allowed: Scientific
Contemporary Mathematics Math 1030 Sample Exam I Chapters 1315 Time Limit: 90 Minutes No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the lefthand margin.
More informationOn the probability scale below mark, with a letter, the probability that the spinner will land
GCSE Exam Questions on Basic Probability. Richard has a box of toy cars. Each car is red or blue or white. 3 of the cars are red. 4 of the cars are blue. of the cars are white. Richard chooses one car
More informationMath 4610, Problems to be Worked in Class
Math 4610, Problems to be Worked in Class Bring this handout to class always! You will need it. If you wish to use an expanded version of this handout with space to write solutions, you can download one
More information2 A fair coin is flipped 8 times. What is the probability of getting more heads than tails? A. 1 2 B E. NOTA
For all questions, answer E. "NOTA" means none of the above answers is correct. Calculator use NO calculators will be permitted on any test other than the Statistics topic test. The word "deck" refers
More informationBell Work. WarmUp Exercises. Two sixsided dice are rolled. Find the probability of each sum or 7
WarmUp Exercises Two sixsided dice are rolled. Find the probability of each sum. 1. 7 Bell Work 2. 5 or 7 3. You toss a coin 3 times. What is the probability of getting 3 heads? WarmUp Notes Exercises
More informationCounting Methods and Probability
CHAPTER Counting Methods and Probability Many good basketball players can make 90% of their free throws. However, the likelihood of a player making several free throws in a row will be less than 90%. You
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More 9.9.3 Practice Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. ) In how many ways can you answer the questions on
More informationSTRAND: PROBABILITY Unit 2 Probability of Two or More Events
STRAND: PROAILITY Unit 2 Probability of Two or More Events TEXT Contents Section 2. Outcome of Two Events 2.2 Probability of Two Events 2. Use of Tree Diagrams 2 Probability of Two or More Events 2. Outcome
More informationConditional probability 2E
Conditional probability 2E 1 a When the first token removed is red, there are 8 tokens remaining in the bag, 4 red and 4 blue. When the first token removed is blue, there are 8 tokens remaining in the
More informationSection 6.5 Conditional Probability
Section 6.5 Conditional Probability Example 1: An urn contains 5 green marbles and 7 black marbles. Two marbles are drawn in succession and without replacement from the urn. a) What is the probability
More informationThe probability setup
CHAPTER 2 The probability setup 2.1. Introduction and basic theory We will have a sample space, denoted S (sometimes Ω) that consists of all possible outcomes. For example, if we roll two dice, the sample
More informationMutually Exclusive Events Algebra 1
Name: Mutually Exclusive Events Algebra 1 Date: Mutually exclusive events are two events which have no outcomes in common. The probability that these two events would occur at the same time is zero. Exercise
More informationMATH 1115, Mathematics for Commerce WINTER 2011 Toby Kenney Homework Sheet 6 Model Solutions
MATH, Mathematics for Commerce WINTER 0 Toby Kenney Homework Sheet Model Solutions. A company has two machines for producing a product. The first machine produces defective products % of the time. The
More information#2. A coin is tossed 40 times and lands on heads 21 times. What is the experimental probability of the coin landing on tails?
1 PreAP Geometry Chapter 14 Test Review Standards/Goals: A.1.f.: I can find the probability of a simple event. F.1.c.: I can use area to solve problems involving geometric probability. S.CP.1: I can define
More informationProbability Rules. 2) The probability, P, of any event ranges from which of the following?
Name: WORKSHEET : Date: Answer the following questions. 1) Probability of event E occurring is... P(E) = Number of ways to get E/Total number of outcomes possible in S, the sample space....if. 2) The probability,
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 1342 Practice Test 2 Ch 4 & 5 Name 1) Nanette must pass through three doors as she walks from her company's foyer to her office. Each of these doors may be locked or unlocked. 1) List the outcomes
More informationA collection of 91 Maths GCSE Sample and Specimen questions from AQA, OCR, PearsonEdexcel and WJEC Eduqas. Name: Total Marks:
Probability 2 (H) A collection of 91 Maths GCSE Sample and Specimen questions from AQA, OCR, PearsonEdexcel and WJEC Eduqas. Name: Total Marks: 1. Andy sometimes gets a lift to and from college. When
More informationKS3 Levels 38. Unit 3 Probability. Homework Booklet. Complete this table indicating the homework you have been set and when it is due by.
Name: Maths Group: Tutor Set: Unit 3 Probability Homework Booklet KS3 Levels 38 Complete this table indicating the homework you have been set and when it is due by. Date Homework Due By Handed In Please
More informationThe probability setup
CHAPTER The probability setup.1. Introduction and basic theory We will have a sample space, denoted S sometimes Ω that consists of all possible outcomes. For example, if we roll two dice, the sample space
More informationCS1802 Week 9: Probability, Expectation, Entropy
CS02 Discrete Structures Recitation Fall 207 October 30  November 3, 207 CS02 Week 9: Probability, Expectation, Entropy Simple Probabilities i. What is the probability that if a die is rolled five times,
More informationProbability and Counting Techniques
Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each
More informationProbability. March 06, J. Boulton MDM 4U1. P(A) = n(a) n(s) Introductory Probability
Most people think they understand odds and probability. Do you? Decision 1: Pick a card Decision 2: Switch or don't Outcomes: Make a tree diagram Do you think you understand probability? Probability Write
More informationPRE TEST. Math in a Cultural Context*
P grade PRE TEST Salmon Fishing: Investigations into A 6P th module in the Math in a Cultural Context* UNIVERSITY OF ALASKA FAIRBANKS Student Name: Grade: Teacher: School: Location of School: Date: *This
More informationFunctional Skills Mathematics
Functional Skills Mathematics Level Learning Resource HD2/L. HD2/L.2 Excellence in skills development Contents HD2/L. Pages 36 HD2/L.2 West Nottinghamshire College 2 HD2/L. HD2/L.2 Information is the
More informationCOMPOUND EVENTS. Judo Math Inc.
COMPOUND EVENTS Judo Math Inc. 7 th grade Statistics Discipline: Black Belt Training Order of Mastery: Compound Events 1. What are compound events? 2. Using organized Lists (7SP8) 3. Using tables (7SP8)
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Statistics Homework Ch 5 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A coin is tossed. Find the probability
More informationIntermediate Math Circles November 1, 2017 Probability I. Problem Set Solutions
Intermediate Math Circles November 1, 2017 Probability I Problem Set Solutions 1. Suppose we draw one card from a wellshuffled deck. Let A be the event that we get a spade, and B be the event we get an
More informationOutcomes: The outcomes of this experiment are yellow, blue, red and green.
(Adapted from http://www.mathgoodies.com/) 1. Sample Space The sample space of an experiment is the set of all possible outcomes of that experiment. The sum of the probabilities of the distinct outcomes
More informationGCSE MATHEMATICS Intermediate Tier, topic sheet. PROBABILITY
GCSE MATHEMATICS Intermediate Tier, topic sheet. PROBABILITY. In a game, a player throws two fair dice, one coloured red the other blue. The score for the throw is the larger of the two numbers showing.
More informationSALES AND MARKETING Department MATHEMATICS. Combinatorics and probabilities. Tutorials and exercises
SALES AND MARKETING Department MATHEMATICS 2 nd Semester Combinatorics and probabilities Tutorials and exercises Online document : http://jffduttc.weebly.com section DUT Maths S2 IUT de SaintEtienne
More information