# TJP TOP TIPS FOR IGCSE STATS & PROBABILITY

Size: px
Start display at page:

Transcription

1 TJP TOP TIPS FOR IGCSE STATS & PROBABILITY Dr T J Price, 2011

2 First, some important words; know what they mean (get someone to test you): Mean the sum of the data values divided by the number of items. The mean of 1, 2, 2, 3, 4, 6 is ( ) 6 = 18 6 = 3 Median the middle data value when all the numbers are listed in order. The median of 1, 2, 2, 3, 4, 6 is (2+3) 2 = 5 2 = 2.5 Mode the most common data value. The mode of 1, 2, 2, 3, 4, 6 is 2 Range the largest data value minus the smallest data value. The range of 1, 2, 2, 3, 4, 6 is 6 1 = 5 Lower Quartile the data value that is one quarter of the way up from the lowest value. The lower quartile of 1, 2, 2, 3, 4, 6 is 2 Upper Quartile the data value that is three quarters of the way up from the lowest value. The upper quartile of 1, 2, 2, 3, 4, 6 is 4 Interquartile Range = Upper Quartile Lower Quartile. The interquartile range of 1, 2, 2, 3, 4, 6 is 4 2 = 2 Frequency the number of times that an event occurs. If a roll a die 20 times and get the number three on four occasions, the frequency is 4. Cumulative Frequency the running total of the frequency values. These running totals are often then plotted as an S-shaped curve. We can then read off the median and the quartiles. Grouped Frequency Table a table where data values are grouped in 'bins'. A survey might record the number of people aged 0-4, 5-9, 10-19, etc. Class Width the width of a 'bin' in a grouped frequency table. Frequency Density the frequency divided by the class width, used in histograms. Histogram a chart plotting frequency density on the y axis with classes along the x axis. It is like a bar chart but corrected for the misleading effect of having differing bin widths. Probability the chance of an event happening. Probability is always a number between 0 (impossible) and 1 (certain). Outcome the result of an event. If a coin is tossed, the possible outcomes are Heads and Tails. Expected Number the number of times you would expect an outcome to occur. If I roll a die 100 times and the chance of rolling a '3' is 0.2, I expect to get 20 '3's. Mutually Exclusive Events events that cannot both/all happen at the same time. If you roll a die, getting a 1 or getting a 2 are mutually exclusive (can't happen together). But having a beard or wearing glasses are not mutually exclusive (can happen together). Independent Events events that do not affect one another's outcomes. If a coin is tossed twice, the outcomes are independent there is no 'memory effect'. But if sweets are picked from a bag and eaten, successive events are not independent. Tree diagram a way of showing all the possible outcomes when two or more events occur, along with their probabilities. The diagram branches repeatedly, like a tree (on its side). Page 2

3 PROBABILITY Probability means the chance of an event happening. It is always given as a number between 0 and 1, with 0 = impossible and 1 = certain. You can give probabilities as decimals or fractions, but never as percentages. The outcome is the result of an event. If you consider all the possible outcomes of an event, their probabilities must add to 1. This is because it is certain that one of the outcomes will happen (we just don't know which one). SKILL: Complete a table of probabilities listing all outcomes. Q: If a spinner is numbered 1, 2, 3, 4 and 5 and it lands on these numbers with the following probabilities, complete the table. Spinner Probability A: Since the probabilities must add up to 1, the missing number is = If the probability of an event happening is p, the probability of it not happening is 1 p. This is because something either happens or it doesn't there's no other option. Events are mutually exclusive if they can't both/all happen at once. An example is getting Heads or getting Tails when you flip a coin; you can't get both. We combine the probabilities of mutually exclusive events by adding them. TJP TOP TIP: Remember ADD-OR (as in 'we add-or statistics'). SKILL: Combine probabilities using the OR rule. Q: For the spinner mentioned in the previous question, find the probability of getting a 2 or a 3. A: Getting a 2 and getting a 3 are mutually exclusive, so we add the probabilities. Prob(getting 2 or 3) = = 0.3. Q: A pupil is picked at random from a class. The probability of picking someone wearing glasses is 0.3 and the probability of picking a girl is 0.5. Explain why the probability of picking a girl or someone wearing glasses is not 0.8. A: Wearing glasses and being a girl are not mutually exclusive; there could be one or more girls who wear glasses. So we can't just add the probabilities; we'd be counting any girls with glasses twice. Page 3

4 Two events A and B are independent if they have no effect on one another. In this case, Prob(A and B) = Prob(A) Prob(B) This is the multiplication law for independent events. SKILL: Combine probabilities using the AND rule. Q: The probability of spinning a 2 is 0.2 and the probability of picking a red ball out of a bag is 0.3. Find the probability of spinning a 2 and picking a red ball. A: These are independent events; they don't affect each other. So we multiply: Prob(2 on spinner and red ball) = = SKILL: List all possible outcomes to solve probability questions. To list all the possible outcomes, you need to be systematic. Here are two common examples: Q: Three fair coins are tossed. List all the possible outcomes. Hence find the probability of getting three tails. A: HHH HHT HTH HTT THH THT TTH TTT There are 8 possible outcomes (spot the pattern...) So the probability of getting TTT is 1/8. Q: Two spinners are marked with numbers from 1 to 4. Draw a table to show all the possible outcomes. If each number is equally likely, find the probability of getting a total of 6. A: x 3.. x. 4. x.. There are 16 possible outcomes (4 4), of which the 3 marked ones add up to 6. So the probability of getting a total of 6 is 3/16. The expected number or expected frequency is the number of times you would expect an event to happen. Simply multiply the probability by the number of trials. SKILL: Find the expected frequency. Q: The probability of getting a '3' when rolling a die is How many times would you expect to get a 3 is you roll it 200 times? A: = 30 times. Page 4

5 SIMPLE TREE DIAGRAMS IGCSE STATS & PROBABILITY Probability problems are often tackled using a tree diagram. This looks like a tree on its side, where the branches show the different outcomes along with their probabilities. The first set of branches from the left correspond to the first event to happen. The next set of branches coming off these correspond to the second event, etc. TJP TOP TIP: Remember MAAD for tree diagrams: Multiply Across, Add Down. This refers to how to combine the probabilities marked on the branches. SKILL: Construct a tree diagram and use it to answer a probability question. Q: The probability of Esmee rolling a six on a fair die is 1/6. a) Draw a tree diagram to show the possible outcomes when she rolls this die twice. b) Use this tree diagram to find the probability of Esmee rolling: i) two sixes ii) exactly one six A: a) Draw the tree diagram. Second roll First roll 1/6 6 1/6 5/6 6 Not 6 5/6 Not 6 1/6 6 5/6 Not 6 b) i) P(two sixes) = ii) P(exactly one six) = = = = = 5 18 [This could be '6' followed by 'not 6', or 'not 6' followed by '6'.] Note: Tree diagrams can have three or more branches at each stage, and three or more stages. But if you end up drawing hundreds of branches, there's probably an easier way... Page 5

6 TREE DIAGRAMS FOR CONDITIONAL PROBABILITY Read the question carefully to see if it says objects are picked without replacement. If so, the probabilities will change after each selection. This is an example of conditional probability, where the probabilities depend on what has already happened. SKILL: Use a tree diagram to answer a conditional probability question. Q: A bag contains 3 stoats and 2 weasels; two animals are then picked at random without replacement. Use a tree diagram to calculate: a) Prob(2 stoats) b) Prob(1 stoat and 1 weasel, in either order) c) Prob(at least one weasel) A: First, draw the tree diagram. 2/4 S 3/5 S 2/4 W 2/5 W 3/4 S 1/4 W [The top right probability is 2/4 because if we remove a stoat, there are now only 2 stoats left out of 4 animals still in the bag.] Now use the tree diagram to answer the questions. a) P(2 stoats) = = 6 20 = 3 10 b) P(1 stoat and 1 weasel) = = = = 3 5 [Note: this could be Stoat then Weasel, or Weasel then Stoat.] c) P(at least one weasel) = 1 P(no weasels) = = = = 7 10 TJP TOP TIP: If it's a 'one of each' question, remember that there is more than one way to get this on your tree diagram. For example, 'A then B' or 'B then A'. Page 6

7 MEAN, MEDIAN, QUARTILES, MODE AND RANGE FROM A LIST Mean = total of all data values total number of items. It's sensitive to any 'freak results' that are unusually high or low. Median = middle data value when sorted into increasing order. If there are two middle data values, take their mean. The median is not sensitive to 'freak results'. Lower Quartile = the median of the bottom half of the list. It's the value ¼ of the way up the list. Upper Quartile = the median of the top half of the list. It's the value ¾ of the way up the list. Interquartile Range = upper quartile lower quartile. It indicates how spread out the data values are. Mode = most common data value. If there are two most common values, the distribution is bimodal. Range = highest value lowest value. SKILL: Find the mean, median, quartiles, mode and range from a list of data. Q: Find the quartiles and median of 4, 5, 6, 8, 10, 13, 15, 16, 19. A: The median is the middle number, 10. The lower quartile is the median of the bottom half 4, 5, 6, 8 which is 5.5. The upper quartile is the median of the top half 13, 15, 16, 19 which is The interquartile range = = 10. [Note: don't include the middle number 10 in the bottom or top half of the list.] Q: Find the mean, median, quartiles, mode and range of 1, 3, 3, 3, 4, 5, 6, 7, 10, 11. A: Mean = ( ) 10 = 5.3 Median = (4+5) 2 = 4.5 Lower Quartile = 3 Upper Quartile = 7 Interquartile Range = 7 3 = 4 Mode = 3 Range = 11 1 = 10 Q: Three numbers are 6, x and 2x (with x>6). Show that the mean is 2 greater than the median. A: The median is x. The mean is (6 + x + 2x) 3 = (6 + 3x) 3 = 2 + x. So the mean (x + 2) is 2 greater than the median (x). Page 7

8 MEAN, MEDIAN, MODE AND RANGE FROM A TABLE To work out these quantities if the data values are listed in a table, do the following. SKILL: Find the mean, median, mode and range from a table. Q: Find the mean, median, mode and range of the following data. length frequency A: Add a column to the table for length frequency. length frequency length frequency Mean = ( ) ( ) = = [Use MAAD Multiply Across, Add Down on the table.] Median = the length category containing the middle (two) items when listed in order. There are 20 items altogether, so we need the 10 th and 11 th items. Count down from the top: Items 1-3 have length 1; Items 4-5 have length 2; Items 6-11 have length 3. So the median is 3. Mode is the length category with the most (the biggest frequency) = 3. Range = biggest length smallest length = 5 1 = 4. TJP TOP TIP: To find the position of the median, do the mean of the first and last positions. So in a list of 123 items, the median is at position ( ) 2 = 62. If this position is 'X and a half', the two middle numbers are at X and X+1. Page 8

9 MEAN, MEDIAN CLASS, MODAL CLASS AND RANGE FROM A GROUPED TABLE If you need to work out these quantities from a grouped table (where data values are grouped into 'bins' so we don't know their exact values any more): Find the mean of grouped data using the middle value of each class. Find the class containing the median (see previous page). The modal class is the group or class with the most (the highest frequency). The range is the upper limit of the highest group (class) minus the lower limit of the lowest group (class). SKILL: Find the mean, median class, modal class and range from a grouped table. Q: Find the mean, median class, modal class and range of the following grouped data. height (cm) frequency A: Add two columns to the table, for the midpoint of the class and for freq midpoint. height (cm) frequency midpoint freq midpoint Mean = = 144 (midpoint MAADness...) This is just an estimate because we don't know the exact data values. Median class = the class containing the middle item, no. 12 in the list. The 12 th item occurs in the class. Modal class = because it has the highest frequency. Range = = 89. Page 9

10 CUMULATIVE FREQUENCY IGCSE STATS & PROBABILITY To find the median and quartiles accurately from grouped data, it is helpful to draw a cumulative frequency graph and read off the values from it. In a cumulative frequency graph, we find the running total of the frequencies. We then plot this against the upper end of each class interval to show how many data values there are up to a particular limit. SKILL: Plot a cumulative frequency curve and find the median and quartiles. Q: Plot a cumulative frequency curve from this table and find the median and quartiles. Value x Frequency f A: First work out the cumulative frequency (running total). Value x Frequency f Cumulative Freq We now plot points at (0, 0) (20, 12) (30, 32) (60, 47) (100, 72) and draw a smooth curve through these points to give that classic S-shaped curve. Cumulative frequency LQ Median UQ x Then we can read off the Median and the Upper and Lower Quartiles off the x-axis. (No actual numbers here this time, but there will be in the exam.) The Interquartile Range = Upper Quartile Lower Quartile. Page 10

11 HISTOGRAMS Histograms are a bit like bar charts except that we must plot frequency density = frequency class width, not the frequency. This means that the area of each bar is equal to the frequency. Also remember that the bars should not have gaps between them. TJP TOP TIP: Two utterly essentially vitally important facts for histograms: Always plot frequency density (= frequency width). Hint: think alphabetically... frequency comes before width. The frequency is given by the area of each bar, not the height. SKILL: Plot a histogram from a grouped frequency table. Q: Display the following data on a histogram. Value x Frequency f A: We must begin by working out the frequency density = frequency width. Draw an extra column (or row) on the table if necessary. Value x Frequency f Freq density = = = = Freq density x TJP TOP TIP: Sometimes they give us a bar that is already drawn on the histogram as well as featuring in the table. Use this known bar to label the y-axis correctly. To read values off the histogram, remember that the area of a bar gives the frequency. Page 11

12 CONTENTS Page Topic 2 Statistics and Probability Words 3-4 Probability 5 Simple Tree Diagrams 6 Tree Diagrams for Conditional Probability 7 Mean, Median, Quartiles, Mode and Range from a List 8 Mean, Median, Mode and Range from a Table 9 Mean, Median Class, Modal Class and Range from a Grouped Table 10 Cumulative Frequency 11 Histograms Page 12

### Algebra I Notes Unit One: Real Number System

Syllabus Objectives: 1.1 The student will organize statistical data through the use of matrices (with and without technology). 1.2 The student will perform addition, subtraction, and scalar multiplication

### Math 146 Statistics for the Health Sciences Additional Exercises on Chapter 3

Math 46 Statistics for the Health Sciences Additional Exercises on Chapter 3 Student Name: Find the indicated probability. ) If you flip a coin three times, the possible outcomes are HHH HHT HTH HTT THH

### FALL 2012 MATH 1324 REVIEW EXAM 4

FALL 01 MATH 134 REVIEW EXAM 4 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the sample space for the given experiment. 1) An ordinary die

### Grade 6 Math Circles Fall Oct 14/15 Probability

1 Faculty of Mathematics Waterloo, Ontario Centre for Education in Mathematics and Computing Grade 6 Math Circles Fall 2014 - Oct 14/15 Probability Probability is the likelihood of an event occurring.

### Numerical: Data with quantity Discrete: whole number answers Example: How many siblings do you have?

Types of data Numerical: Data with quantity Discrete: whole number answers Example: How many siblings do you have? Continuous: Answers can fall anywhere in between two whole numbers. Usually any type of

### Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics of Chance Free-Response 1. A spinner with regions numbered 1 to 4 is spun and a coin is tossed. Both the number spun and whether the coin lands heads or tails is

### STAT 155 Introductory Statistics. Lecture 11: Randomness and Probability Model

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 11: Randomness and Probability Model 10/5/06 Lecture 11 1 The Monty Hall Problem Let s Make A Deal: a game show

### Algebra 2 P49 Pre 10 1 Measures of Central Tendency Box and Whisker Plots Variation and Outliers

Algebra 2 P49 Pre 10 1 Measures of Central Tendency Box and Whisker Plots Variation and Outliers 10 1 Sample Spaces and Probability Mean Average = 40/8 = 5 Measures of Central Tendency 2,3,3,4,5,6,8,9

### Name Class Date. Introducing Probability Distributions

Name Class Date Binomial Distributions Extension: Distributions Essential question: What is a probability distribution and how is it displayed? 8-6 CC.9 2.S.MD.5(+) ENGAGE Introducing Distributions Video

### Probability --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..

### PROBABILITY 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: 08-10-2015 Mathematics Revision Guides Probability

### STOR 155 Introductory Statistics. Lecture 10: Randomness and Probability Model

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 10: Randomness and Probability Model 10/6/09 Lecture 10 1 The Monty Hall Problem Let s Make A Deal: a game show

### heads 1/2 1/6 roll a die sum on 2 dice 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 1, 2, 3, 4, 5, 6 heads tails 3/36 = 1/12 toss a coin trial: an occurrence

trial: an occurrence roll a die toss a coin sum on 2 dice sample space: all the things that could happen in each trial 1, 2, 3, 4, 5, 6 heads tails 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 example of an outcome:

### Probability of Independent and Dependent Events

706 Practice A Probability of In and ependent Events ecide whether each set of events is or. Explain your answer.. A student spins a spinner and rolls a number cube.. A student picks a raffle ticket from

### 1) What is the total area under the curve? 1) 2) What is the mean of the distribution? 2)

Math 1090 Test 2 Review Worksheet Ch5 and Ch 6 Name Use the following distribution to answer the question. 1) What is the total area under the curve? 1) 2) What is the mean of the distribution? 2) 3) Estimate

### Probability. 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

### Probability Assignment

Name Probability Assignment Student # Hr 1. An experiment consists of spinning the spinner one time. a. How many possible outcomes are there? b. List the sample space for the experiment. c. Determine the

### Probability. Ms. Weinstein Probability & Statistics

Probability Ms. Weinstein Probability & Statistics Definitions Sample Space The sample space, S, of a random phenomenon is the set of all possible outcomes. Event An event is a set of outcomes of a random

### Name: Class: Date: Probability/Counting Multiple Choice Pre-Test

Name: _ lass: _ ate: Probability/ounting Multiple hoice Pre-Test Multiple hoice Identify the choice that best completes the statement or answers the question. 1 The dartboard has 8 sections of equal area.

### Midterm 2 Practice Problems

Midterm 2 Practice Problems May 13, 2012 Note that these questions are not intended to form a practice exam. They don t necessarily cover all of the material, or weight the material as I would. They are

### Statistics 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

### There is no class tomorrow! Have a good weekend! Scores will be posted in Compass early Friday morning J

STATISTICS 100 EXAM 3 Fall 2016 PRINT NAME (Last name) (First name) *NETID CIRCLE SECTION: L1 12:30pm L2 3:30pm Online MWF 12pm Write answers in appropriate blanks. When no blanks are provided CIRCLE your

### Math. Integrated. Trimester 3 Revision Grade 7. Zayed Al Thani School. ministry of education.

ministry of education Department of Education and Knowledge Zayed Al Thani School www.z2school.com Integrated Math Grade 7 2017-2018 Trimester 3 Revision الوزارة كتاب عن تغني ال المراجعة هذه 0 Ministry

### a) Getting 10 +/- 2 head in 20 tosses is the same probability as getting +/- heads in 320 tosses

Question 1 pertains to tossing a fair coin (8 pts.) Fill in the blanks with the correct numbers to make the 2 scenarios equally likely: a) Getting 10 +/- 2 head in 20 tosses is the same probability as

### Mathematicsisliketravellingona rollercoaster.sometimesyouron. Mathematics. ahighothertimesyouronalow.ma keuseofmathsroomswhenyouro

Mathematicsisliketravellingona rollercoaster.sometimesyouron Mathematics ahighothertimesyouronalow.ma keuseofmathsroomswhenyouro Stage 6 nalowandshareyourpracticewit Handling Data hotherswhenonahigh.successwi

### Page 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.

### EECS 203 Spring 2016 Lecture 15 Page 1 of 6

EECS 203 Spring 2016 Lecture 15 Page 1 of 6 Counting We ve been working on counting for the last two lectures. We re going to continue on counting and probability for about 1.5 more lectures (including

### Probability. Dr. Zhang Fordham Univ.

Probability! Dr. Zhang Fordham Univ. 1 Probability: outline Introduction! Experiment, event, sample space! Probability of events! Calculate Probability! Through counting! Sum rule and general sum rule!

### Math 147 Elementary Probability/Statistics I Additional Exercises on Chapter 4: Probability

Math 147 Elementary Probability/Statistics I Additional Exercises on Chapter 4: Probability Student Name: Find the indicated probability. 1) If you flip a coin three times, the possible outcomes are HHH

### Part 1: I can express probability as a fraction, decimal, and percent

Name: Pattern: Part 1: I can express probability as a fraction, decimal, and percent For #1 to #4, state the probability of each outcome. Write each answer as a) a fraction b) a decimal c) a percent Example:

### UNIT 4 APPLICATIONS OF PROBABILITY Lesson 1: Events. Instruction. Guided Practice Example 1

Guided Practice Example 1 Bobbi tosses a coin 3 times. What is the probability that she gets exactly 2 heads? Write your answer as a fraction, as a decimal, and as a percent. Sample space = {HHH, HHT,

### Raise your hand if you rode a bus within the past month. Record the number of raised hands.

166 CHAPTER 3 PROBABILITY TOPICS Raise your hand if you rode a bus within the past month. Record the number of raised hands. Raise your hand if you answered "yes" to BOTH of the first two questions. Record

### Lesson 10: Using Simulation to Estimate a Probability

Lesson 10: Using Simulation to Estimate a Probability Classwork In previous lessons, you estimated probabilities of events by collecting data empirically or by establishing a theoretical probability model.

### 1. 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

### b. 2 ; the probability of choosing a white d. P(white) 25, or a a. Since the probability of choosing a

Applications. a. P(green) =, P(yellow) = 2, or 2, P(red) = 2 ; three of the four blocks are not red. d. 2. a. P(green) = 2 25, P(purple) = 6 25, P(orange) = 2 25, P(yellow) = 5 25, or 5 2 6 2 5 25 25 25

### Probability Simulation User s Manual

Probability Simulation User s Manual Documentation of features and usage for Probability Simulation Copyright 2000 Corey Taylor and Rusty Wagner 1 Table of Contents 1. General Setup 3 2. Coin Section 4

### Statistics 101: Section L Laboratory 10

Statistics 101: Section L Laboratory 10 This lab looks at the sampling distribution of the sample proportion pˆ and probabilities associated with sampling from a population with a categorical variable.

### Independence Is The Word

Problem 1 Simulating Independent Events Describe two different events that are independent. Describe two different events that are not independent. The probability of obtaining a tail with a coin toss

### , 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

### NumberSense Companion Workbook Grade 4

NumberSense Companion Workbook Grade 4 Sample Pages (ENGLISH) Working in the NumberSense Companion Workbook The NumberSense Companion Workbooks address measurement, spatial reasoning (geometry) and data

### This unit will help you work out probability and use experimental probability and frequency trees. Key points

Get started Probability This unit will help you work out probability and use experimental probability and frequency trees. AO Fluency check There are 0 marbles in a bag. 9 of the marbles are red, 7 are

### In how many ways can the letters of SEA be arranged? In how many ways can the letters of SEE be arranged?

-Pick up Quiz Review Handout by door -Turn to Packet p. 5-6 In how many ways can the letters of SEA be arranged? In how many ways can the letters of SEE be arranged? - Take Out Yesterday s Notes we ll

### THOMAS WHITHAM SIXTH FORM

THOMAS WHITHAM SIXTH FORM Handling Data Levels 6 8 S. J. Cooper Probability Tree diagrams & Sample spaces Statistical Graphs Scatter diagrams Mean, Mode & Median Year 9 B U R N L E Y C A M P U S, B U R

### Homework #1-19: Use the Counting Principle to answer the following questions.

Section 4.3: Tree Diagrams and the Counting Principle Homework #1-19: Use the Counting Principle to answer the following questions. 1) If two dates are selected at random from the 365 days of the year

### RANDOM EXPERIMENTS AND EVENTS

Random Experiments and Events 18 RANDOM EXPERIMENTS AND EVENTS In day-to-day 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

### Counting methods (Part 4): More combinations

April 13, 2009 Counting methods (Part 4): More combinations page 1 Counting methods (Part 4): More combinations Recap of last lesson: The combination number n C r is the answer to this counting question:

### 3. a. P(white) =, or. b. ; the probability of choosing a white block. d. P(white) =, or. 4. a. = 1 b. 0 c. = 0

Answers Investigation ACE Assignment Choices Problem. Core, 6 Other Connections, Extensions Problem. Core 6 Other Connections 7 ; unassigned choices from previous problems Problem. Core 7 9 Other Connections

### @Holyheadmaths GCSE METHODS REVISION- MARCH Higher Paper 1 (Non calculator)

@Holyheadmaths GCSE METHODS REVISION- MARCH 201 Higher Paper 1 (Non calculator) Adding Fractions To add fractions we need to multiply across the bottom and then across the fractions- flat across your bottom

### (Notice that the mean doesn t have to be a whole number and isn t normally part of the original set of data.)

One-Variable Statistics Descriptive statistics that analyze one characteristic of one sample Where s the middle? How spread out is it? Where do different pieces of data compare? To find 1-variable statistics

### Section 1.5 Graphs and Describing Distributions

Section 1.5 Graphs and Describing Distributions Data can be displayed using graphs. Some of the most common graphs used in statistics are: Bar graph Pie Chart Dot plot Histogram Stem and leaf plot Box

### When a number cube is rolled once, the possible numbers that could show face up are

C3 Chapter 12 Understanding Probability Essential question: How can you describe the likelihood of an event? Example 1 Likelihood of an Event When a number cube is rolled once, the possible numbers that

### Chapter 4. Displaying and Summarizing Quantitative Data. Copyright 2012, 2008, 2005 Pearson Education, Inc.

Chapter 4 Displaying and Summarizing Quantitative Data Copyright 2012, 2008, 2005 Pearson Education, Inc. Dealing With a Lot of Numbers Summarizing the data will help us when we look at large sets of quantitative

### Probability Exercise 2

Probability Exercise 2 1 Question 9 A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will

### UNIT 5: RATIO, PROPORTION, AND PERCENT WEEK 20: Student Packet

Name Period Date UNIT 5: RATIO, PROPORTION, AND PERCENT WEEK 20: Student Packet 20.1 Solving Proportions 1 Add, subtract, multiply, and divide rational numbers. Use rates and proportions to solve problems.

### Paper 1. Calculator not allowed. Mathematics test. First name. Last name. School. Remember KEY STAGE 3 TIER 6 8

Ma KEY STAGE 3 Mathematics test TIER 6 8 Paper 1 Calculator not allowed First name Last name School 2007 Remember The test is 1 hour long. You must not use a calculator for any question in this test. You

### c. If you roll the die six times what are your chances of getting at least one d. roll.

1. Find the area under the normal curve: a. To the right of 1.25 (100-78.87)/2=10.565 b. To the left of -0.40 (100-31.08)/2=34.46 c. To the left of 0.80 (100-57.63)/2=21.185 d. Between 0.40 and 1.30 for

### STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes

STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes Pengyuan (Penelope) Wang May 25, 2011 Review We have discussed counting techniques in Chapter 1. (Principle

### PASS Sample Size Software

Chapter 945 Introduction This section describes the options that are available for the appearance of a histogram. A set of all these options can be stored as a template file which can be retrieved later.

### Year 9 Unit G Revision. Exercise A

Year 9 Unit G Revision Exercise A 1.) Find the mode, median, mean, range and interquartile range of each of the following lists. a.) 11, 13, 13, 16, 16, 17, 19, 20, 24, 24, 24, 25, 30 b.) 21, 36, 78, 45,

### Probability Interactives from Spire Maths A Spire Maths Activity

Probability Interactives from Spire Maths A Spire Maths Activity https://spiremaths.co.uk/ia/ There are 12 sets of Probability Interactives: each contains a main and plenary flash file. Titles are shown

### Lesson 4: Calculating Probabilities for Chance Experiments with Equally Likely Outcomes

Lesson : Calculating Probabilities for Chance Experiments with Equally Likely Outcomes Classwork Example : heoretical Probability In a previous lesson, you saw that to find an estimate of the probability

### CHAPTER 6 PROBABILITY. Chapter 5 introduced the concepts of z scores and the normal curve. This chapter takes

CHAPTER 6 PROBABILITY Chapter 5 introduced the concepts of z scores and the normal curve. This chapter takes these two concepts a step further and explains their relationship with another statistical concept

### Use a tree diagram to find the number of possible outcomes. 2. How many outcomes are there altogether? 2.

Use a tree diagram to find the number of possible outcomes. 1. A pouch contains a blue chip and a red chip. A second pouch contains two blue chips and a red chip. A chip is picked from each pouch. The

### Name: Date: Period: Histogram Worksheet

Name: Date: Period: Histogram Worksheet 1 5. For the following five histograms, list at least 3 characteristics that describe each histogram (consider symmetric, skewed to left, skewed to right, unimodal,

### 3.6 Theoretical and Experimental Coin Tosses

wwwck12org Chapter 3 Introduction to Discrete Random Variables 36 Theoretical and Experimental Coin Tosses Here you ll simulate coin tosses using technology to calculate experimental probability Then you

### Probability 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,

### Chapter 4: Introduction to Probability

MTH 243 Chapter 4: Introduction to Probability Suppose that we found that one of our pieces of data was unusual. For example suppose our pack of M&M s only had 30 and that was 3.1 standard deviations below

### Class 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:

### DESCRIBING DATA. Frequency Tables, Frequency Distributions, and Graphic Presentation

DESCRIBING DATA Frequency Tables, Frequency Distributions, and Graphic Presentation Raw Data A raw data is the data obtained before it is being processed or arranged. 2 Example: Raw Score A raw score is

### Year 9 mathematics: holiday revision. 2 How many nines are there in fifty-four?

DAY 1 ANSWERS Mental questions 1 Multiply seven by seven. 49 2 How many nines are there in fifty-four? 54 9 = 6 6 3 What number should you add to negative three to get the answer five? -3 0 5 8 4 Add two

### Core Connections, Course 2 Checkpoint Materials

Core Connections, Course Checkpoint Materials Notes to Students (and their Teachers) Students master different skills at different speeds. No two students learn exactly the same way at the same time. At

### PROBABILITY. 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

### Grade 8 Math Assignment: Probability

Grade 8 Math Assignment: Probability Part 1: Rock, Paper, Scissors - The Study of Chance Purpose An introduction of the basic information on probability and statistics Materials: Two sets of hands Paper

### MULTIPLE 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

### Mini-Unit. Data & Statistics. Investigation 1: Correlations and Probability in Data

Mini-Unit Data & Statistics Investigation 1: Correlations and Probability in Data I can Measure Variation in Data and Strength of Association in Two-Variable Data Lesson 3: Probability Probability is a

### Paper 1. Calculator not allowed. Mathematics test. First name. Last name. School. Remember KEY STAGE 3 TIER 5 7

Ma KEY STAGE 3 Mathematics test TIER 5 7 Paper 1 Calculator not allowed First name Last name School 2007 Remember The test is 1 hour long. You must not use a calculator for any question in this test. You

### MATH CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #1 - SPRING DR. DAVID BRIDGE

MATH 205 - CALCULUS & STATISTICS/BUSN - PRACTICE EXAM # - SPRING 2006 - DR. DAVID BRIDGE TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. Tell whether the statement is

### Unit 11 Probability. Round 1 Round 2 Round 3 Round 4

Study Notes 11.1 Intro to Probability Unit 11 Probability Many events can t be predicted with total certainty. The best thing we can do is say how likely they are to happen, using the idea of probability.

### Heads Up! A c t i v i t y 5. The Problem. Name Date

. Name Date A c t i v i t y 5 Heads Up! In this activity, you will study some important concepts in a branch of mathematics known as probability. You are using probability when you say things like: It

### Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5 {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)}

Section 8: Random Variables and probability distributions of discrete random variables In the previous sections we saw that when we have numerical data, we can calculate descriptive statistics such as

### She concludes that the dice is biased because she expected to get only one 6. Do you agree with June's conclusion? Briefly justify your answer.

PROBABILITY & STATISTICS TEST Name: 1. June suspects that a dice may be biased. To test her suspicions, she rolls the dice 6 times and rolls 6, 6, 4, 2, 6, 6. She concludes that the dice is biased because

### Date. Probability. Chapter

Date Probability Contests, lotteries, and games offer the chance to win just about anything. You can win a cup of coffee. Even better, you can win cars, houses, vacations, or millions of dollars. Games

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Practice for Final Exam Name Identify the following variable as either qualitative or quantitative and explain why. 1) The number of people on a jury A) Qualitative because it is not a measurement or a

### episteme Probability

episteme Probability Problem Set 3 Please use CAPITAL letters FIRST NAME LAST NAME SCHOOL CLASS DATE / / Set 3 1 episteme, 2010 Set 3 2 episteme, 2010 Coin A fair coin is one which is equally likely to

### b) Find the exact probability of seeing both heads and tails in three tosses of a fair coin. (Theoretical Probability)

Math 1351 Activity 2(Chapter 11)(Due by EOC Mar. 26) Group # 1. A fair coin is tossed three times, and we would like to know the probability of getting both a heads and tails to occur. Here are the results

### MEP 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)

### Diamond ( ) (Black coloured) (Black coloured) (Red coloured) ILLUSTRATIVE EXAMPLES

CHAPTER 15 PROBABILITY Points to Remember : 1. In the experimental approach to probability, we find the probability of the occurence of an event by actually performing the experiment a number of times

### = = 0.1%. On the other hand, if there are three winning tickets, then the probability of winning one of these winning tickets must be 3 (1)

MA 5 Lecture - Binomial Probabilities Wednesday, April 25, 202. Objectives: Introduce combinations and Pascal s triangle. The Fibonacci sequence had a number pattern that we could analyze in different

### Math 147 Lecture Notes: Lecture 21

Math 147 Lecture Notes: Lecture 21 Walter Carlip March, 2018 The Probability of an Event is greater or less, according to the number of Chances by which it may happen, compared with the whole number of

### Probability and Counting Rules. Chapter 3

Probability and Counting Rules Chapter 3 Probability as a general concept can be defined as the chance of an event occurring. Many people are familiar with probability from observing or playing games of

### Probability: Part 1 1/28/16

Probability: Part 1 1/28/16 The Kind of Studies We Can t Do Anymore Negative operant conditioning with a random reward system Addictive behavior under a random reward system FBJ murine osteosarcoma viral

### CHAPTER 2 PROBABILITY. 2.1 Sample Space. 2.2 Events

CHAPTER 2 PROBABILITY 2.1 Sample Space A probability model consists of the sample space and the way to assign probabilities. Sample space & sample point The sample space S, is the set of all possible outcomes

### Section Theoretical and Experimental Probability...Wks 3

Name: Class: Date: Section 6.8......Theoretical and Experimental Probability...Wks 3. Eight balls numbered from to 8 are placed in a basket. One ball is selected at random. Find the probability that it

### Exercise 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

### If a fair coin is tossed 10 times, what will we see? 24.61% 20.51% 20.51% 11.72% 11.72% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098%

Coin tosses If a fair coin is tossed 10 times, what will we see? 30% 25% 24.61% 20% 15% 10% Probability 20.51% 20.51% 11.72% 11.72% 5% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098% 0 1 2 3 4 5 6 7 8 9 10 Number

### 12 Probability. Introduction Randomness

2 Probability Assessment statements 5.2 Concepts of trial, outcome, equally likely outcomes, sample space (U) and event. The probability of an event A as P(A) 5 n(a)/n(u ). The complementary events as

### ECON 214 Elements of Statistics for Economists

ECON 214 Elements of Statistics for Economists Session 4 Probability Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education School of Continuing