# It is remarkable that a science, which began with the consideration of games of chance, should be elevated to the rank of the most important

Save this PDF as:

Size: px
Start display at page:

Download "It is remarkable that a science, which began with the consideration of games of chance, should be elevated to the rank of the most important"

## Transcription

1 PROBABILLITY 271 PROBABILITY CHAPTER 15 It is remarkable that a science, which began with the consideration of games of chance, should be elevated to the rank of the most important subject of human knowledge. Pierre Simon Laplace 15.1 Introduction In everyday life, we come across statements such as (1) It will probably rain today. (2) I doubt that he will pass the test. (3) Most probably, Kavita will stand first in the annual examination. (4) Chances are high that the prices of diesel will go up. (5) There is a chance of India winning a toss in today s match. The words probably, doubt, most probably, chances, etc., used in the statements above involve an element of uncertainty. For example, in (1), probably rain will mean it may rain or may not rain today. We are predicting rain today based on our past experience when it rained under similar conditions. Similar predictions are also made in other cases listed in (2) to (5). The uncertainty of probably etc can be measured numerically by means of probability in many cases. Though probability started with gambling, it has been used extensively in the fields of Physical Sciences, Commerce, Biological Sciences, Medical Sciences, Weather Forecasting, etc.

2 272 MATHEMATICS 15.2 Probability an Experimental Approach Blaise Pascal ( ) The concept of probability developed in a very strange manner. In 1654, a gambler Chevalier de Mere, approached the well-known 17th century French philosopher and mathematician Blaise Pascal regarding certain dice problems. Pascal became interested in these problems, studied them and discussed them with another French mathematician, Pierre de Fermat. Both Pascal and Fermat solved the problems independently. This work was the beginning of Probability Theory. Fig Fig The first book on the subject was written by the Italian mathematician, J.Cardan ( ). The title of the book was Book on Games of Chance (Liber de Ludo Aleae), published in Notable contributions were also made by mathematicians J. Bernoulli ( ), P. Laplace ( ), A.A. Markov ( ) and A.N. Kolmogorov (born 1903). In earlier classes, you have had a glimpse of probability when you performed experiments like tossing of coins, throwing of dice, etc., and observed their outcomes. You will now learn to measure the chance of occurrence of a particular outcome in an experiment. Activity 1 : (i) Take any coin, toss it ten times and note down the number of times a head and a tail come up. Record your observations in the form of the following table Table 15.1 Number of times Number of times Number of times the coin is tossed head comes up tail comes up 10 Write down the values of the following fractions: Pierre de Fermat ( ) Number of times a head comes up Total number of times the coin is tossed and Number of times a tail comes up Total number of times the coin is tossed

3 PROBABILLITY 273 (ii) Toss the coin twenty times and in the same way record your observations as above. Again find the values of the fractions given above for this collection of observations. (iii) Repeat the same experiment by increasing the number of tosses and record the number of heads and tails. Then find the values of the corresponding fractions. You will find that as the number of tosses gets larger, the values of the fractions come closer to 0.5. To record what happens in more and more tosses, the following group activity can also be performed: Acitivity 2 : Divide the class into groups of 2 or 3 students. Let a student in each group toss a coin 15 times. Another student in each group should record the observations regarding heads and tails. [Note that coins of the same denomination should be used in all the groups. It will be treated as if only one coin has been tossed by all the groups.] Now, on the blackboard, make a table like Table First, Group 1 can write down its observations and calculate the resulting fractions. Then Group 2 can write down its observations, but will calculate the fractions for the combined data of Groups 1 and 2, and so on. (We may call these fractions as cumulative fractions.) We have noted the first three rows based on the observations given by one class of students. Table 15.2 Group Number Number Cumulative number of heads Cumulative number of tails of of Total number of times Total number of times heads tails the coin is tossed the coin is tossed (1) (2) (3) (4) (5) = = = = What do you observe in the table? You will find that as the total number of tosses of the coin increases, the values of the fractions in Columns (4) and (5) come nearer and nearer to 0.5. Activity 3 : (i) Throw a die* 20 times and note down the number of times the numbers *A die is a well balanced cube with its six faces marked with numbers from 1 to 6, one number on one face. Sometimes dots appear in place of numbers.

4 274 MATHEMATICS 1, 2, 3, 4, 5, 6 come up. Record your observations in the form of a table, as in Table 15.3: Table 15.3 Number of times a die is thrown Number of times these scores turn up Find the values of the following fractions: Number of times 1 turned up Total number of times the die is thrown Number of times 2 turned up Total number of times the die is thrown Number of times 6 turned up Total number of times the die is thrown (ii) Now throw the die 40 times, record the observations and calculate the fractions as done in (i). As the number of throws of the die increases, you will find that the value of each fraction calculated in (i) and (ii) comes closer and closer to 1 6. To see this, you could perform a group activity, as done in Activity 2. Divide the students in your class, into small groups. One student in each group should throw a die ten times. Observations should be noted and cumulative fractions should be calculated. The values of the fractions for the number 1 can be recorded in Table This table can be extended to write down fractions for the other numbers also or other tables of the same kind can be created for the other numbers.

5 PROBABILLITY 275 Table 15.4 Group Total number of times a die Cumulative number of times 1 turned up is thrown in a group Total number of times the die is thrown (1) (2) (3) The dice used in all the groups should be almost the same in size and appearence. Then all the throws will be treated as throws of the same die. What do you observe in these tables? You will find that as the total number of throws gets larger, the fractions in Column (3) move closer and closer to 1 6. Activity 4 : (i) Toss two coins simultaneously ten times and record your observations in the form of a table as given below: Table 15.5 Number of times the Number of times Number of times Number of times two coins are tossed no head comes up one head comes up two heads come up 10 Write down the fractions: A = B = C = Number of times no head comes up Total number of times two coins are tossed Number of times one head comes up Total number of times two coins are tossed Number of times two heads come up Total number of times two coins are tossed Calculate the values of these fractions.

6 276 MATHEMATICS Now increase the number of tosses (as in Activitiy 2). You will find that the more the number of tosses, the closer are the values of A, B and C to 0.25, 0.5 and 0.25, respectively. In Activity 1, each toss of a coin is called a trial. Similarly in Activity 3, each throw of a die is a trial, and each simultaneous toss of two coins in Activity 4 is also a trial. So, a trial is an action which results in one or several outcomes. The possible outcomes in Activity 1 were Head and Tail; whereas in Activity 3, the possible outcomes were 1, 2, 3, 4, 5 and 6. In Activity 1, the getting of a head in a particular throw is an event with outcome head. Similarly, getting a tail is an event with outcome tail. In Activity 2, the getting of a particular number, say 1, is an event with outcome 1. If our experiment was to throw the die for getting an even number, then the event would consist of three outcomes, namely, 2, 4 and 6. So, an event for an experiment is the collection of some outcomes of the experiment. In Class X, you will study a more formal definition of an event. So, can you now tell what the events are in Activity 4? With this background, let us now see what probability is. Based on what we directly observe as the outcomes of our trials, we find the experimental or empirical probability. Let n be the total number of trials. The empirical probability P(E) of an event E happening, is given by Number of trials in which the event happened P(E) = The total number of trials In this chapter, we shall be finding the empirical probability, though we will write probability for convenience. Let us consider some examples. To start with let us go back to Activity 2, and Table In Column (4) of this table, what is the fraction that you calculated? Nothing, but it is the empirical probability of getting a head. Note that this probability kept changing depending on the number of trials and the number of heads obtained in these trials. Similarly, the empirical probability of getting a tail is obtained in Column (5) of Table This is 12 to start with, then 15 it is 2 28, then, and so on So, the empirical probability depends on the number of trials undertaken, and the number of times the outcomes you are looking for coming up in these trials.

7 PROBABILLITY 277 Activity 5 : Before going further, look at the tables you drew up while doing Activity 3. Find the probabilities of getting a 3 when throwing a die a certain number of times. Also, show how it changes as the number of trials increases. Now let us consider some other examples. Example 1 : A coin is tossed 1000 times with the following frequencies: Head : 455, Tail : 545 Compute the probability for each event. Solution : Since the coin is tossed 1000 times, the total number of trials is Let us call the events of getting a head and of getting a tail as E and F, respectively. Then, the number of times E happens, i.e., the number of times a head come up, is 455. Number of heads So, the probability of E = Total number of trials i.e., P (E) = = Number of tails Similarly, the probability of the event of getting a tail = Total number of trials i.e., P(F) = = Note that in the example above, P(E) + P(F) = = 1, and E and F are the only two possible outcomes of each trial. Example 2 : Two coins are tossed simultaneously 500 times, and we get Two heads : 105 times One head : 275 times No head : 120 times Find the probability of occurrence of each of these events. Solution : Let us denote the events of getting two heads, one head and no head by E 1, E 2 and E 3, respectively. So, P(E 1 )= = 0.21 P(E 2 )= = 0.55 P(E 3 )= = 0.24

8 278 MATHEMATICS Observe that P(E 1 ) + P(E 2 ) + P(E 3 ) = 1. Also E 1, E 2 and E 3 cover all the outcomes of a trial. Example 3 : A die is thrown 1000 times with the frequencies for the outcomes 1, 2, 3, 4, 5 and 6 as given in the following table : Table 15.6 Outcome Frequency Find the probability of getting each outcome. Solution : Let E i denote the event of getting the outcome i, where i = 1, 2, 3, 4, 5, 6. Then Frequency of 1 Probability of the outcome 1 = P(E 1 ) = Total number of times the die is thrown = = Similarly, P(E 2 ) = = 0.15, P(E 3 ) = = 0.157, P(E 4 ) = = 0.149, P(E 5 ) = = and P(E 6 ) = = Note that P(E 1 ) + P(E 2 ) + P(E 3 ) + P(E 4 ) + P(E 5 ) + P(E 6 ) = 1 Also note that: (i) The probability of each event lies between 0 and 1. (ii) The sum of all the probabilities is 1. (iii) E 1, E 2,..., E 6 cover all the possible outcomes of a trial. Example 4 : On one page of a telephone directory, there were 200 telephone numbers. The frequency distribution of their unit place digit (for example, in the number , the unit place digit is 3) is given in Table 15.7 :

9 PROBABILLITY 279 Table 15.7 Digit Frequency Without looking at the page, the pencil is placed on one of these numbers, i.e., the number is chosen at random. What is the probability that the digit in its unit place is 6? Solution : The probability of digit 6 being in the unit place Frequency of 6 = Total number of selected telephone numbers = = 0.07 You can similarly obtain the empirical probabilities of the occurrence of the numbers having the other digits in the unit place. Example 5 : The record of a weather station shows that out of the past 250 consecutive days, its weather forecasts were correct 175 times. (i) What is the probability that on a given day it was correct? (ii) What is the probability that it was not correct on a given day? Solution : The total number of days for which the record is available = 250 (i) P(the forecast was correct on a given day) Number of days when the forecast was correct = Total number of days for which the record is available = = 0.7 (ii) The number of days when the forecast was not correct = = 75 So, P(the forecast was not correct on a given day) = = 0.3 Notice that: P(forecast was correct on a given day) + P(forecast was not correct on a given day) = = 1

10 280 MATHEMATICS Example 6 : A tyre manufacturing company kept a record of the distance covered before a tyre needed to be replaced. The table shows the results of 1000 cases. Table 15.8 Distance (in km) less than to to more than Frequency If you buy a tyre of this company, what is the probability that : (i) it will need to be replaced before it has covered 4000 km? (ii) it will last more than 9000 km? (iii) it will need to be replaced after it has covered somewhere between 4000 km and km? Solution : (i) The total number of trials = The frequency of a tyre that needs to be replaced before it covers 4000 km is So, P(tyre to be replaced before it covers 4000 km) = 1000 = 0.02 (ii) The frequency of a tyre that will last more than 9000 km is = 770 So, P(tyre will last more than 9000 km) = = 0.77 (iii) The frequency of a tyre that requires replacement between 4000 km and km is = 535. So, P(tyre requiring replacement between 4000 km and km) = = Example 7 : The percentage of marks obtained by a student in the monthly unit tests are given below: Table 15.9 Unit test I II III IV V Percentage of marks obtained Based on this data, find the probability that the student gets more than 70% marks in a unit test.

11 PROBABILLITY 281 Solution : The total number of unit tests held is 5. The number of unit tests in which the student obtained more than 70% marks is 3. So, P(scoring more than 70% marks) = 3 5 = 0.6 Example 8 : An insurance company selected 2000 drivers at random (i.e., without any preference of one driver over another) in a particular city to find a relationship between age and accidents. The data obtained are given in the following table: Age of drivers (in years) Table Accidents in one year over Above Find the probabilities of the following events for a driver chosen at random from the city: (i) being years of age and having exactly 3 accidents in one year. (ii) being years of age and having one or more accidents in a year. (iii) having no accidents in one year. Solution : Total number of drivers = (i) The number of drivers who are years old and have exactly 3 accidents in one year is So, P (driver is years old with exactly 3 accidents) = 2000 = (ii) The number of drivers years of age and having one or more accidents in one year = = 225 So, P(driver is years of age and having one or more accidents) = 225 = (iii) The number of drivers having no accidents in one year = = 1305

12 282 MATHEMATICS Therefore, P(drivers with no accident) = = Example 9 : Consider the frequency distribution table (Table 14.3, Example 4, Chapter 14), which gives the weights of 38 students of a class. (i) Find the probability that the weight of a student in the class lies in the interval kg. (ii) Give two events in this context, one having probability 0 and the other having probability 1. Solution : (i) The total number of students is 38, and the number of students with weight in the interval kg is 3. So, P(weight of a student is in the interval kg) = 3 38 = (ii) For instance, consider the event that a student weighs 30 kg. Since no student has this weight, the probability of occurrence of this event is 0. Similarly, the probability of a student weighing more than 30 kg is = 1. Example 10 : Fifty seeds were selected at random from each of 5 bags of seeds, and were kept under standardised conditions favourable to germination. After 20 days, the number of seeds which had germinated in each collection were counted and recorded as follows: Table Bag Number of seeds germinated What is the probability of germination of (i) more than 40 seeds in a bag? (ii) 49 seeds in a bag? (iii) more that 35 seeds in a bag? Solution : Total number of bags is 5. (i) Number of bags in which more than 40 seeds germinated out of 50 seeds is 3. P(germination of more than 40 seeds in a bag) = 5 3 = 0.6

13 PROBABILLITY 283 (ii) Number of bags in which 49 seeds germinated = 0. P(germination of 49 seeds in a bag) = 0 5 = 0. (iii) Number of bags in which more than 35 seeds germinated = 5. So, the required probability = 5 5 = 1. Remark : In all the examples above, you would have noted that the probability of an event can be any fraction from 0 to 1. EXERCISE In a cricket match, a batswoman hits a boundary 6 times out of 30 balls she plays. Find the probability that she did not hit a boundary families with 2 children were selected randomly, and the following data were recorded: Number of girls in a family Number of families Compute the probability of a family, chosen at random, having (i) 2 girls (ii) 1 girl (iii) No girl Also check whether the sum of these probabilities is Refer to Example 5, Section 14.4, Chapter 14. Find the probability that a student of the class was born in August. 4. Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes: Outcome 3 heads 2 heads 1 head No head Frequency If the three coins are simultaneously tossed again, compute the probability of 2 heads coming up. 5. An organisation selected 2400 families at random and surveyed them to determine a relationship between income level and the number of vehicles in a family. The

14 284 MATHEMATICS information gathered is listed in the table below: Monthly income Vehicles per family (in Rs) Above 2 Less than or more Suppose a family is chosen. Find the probability that the family chosen is (i) earning Rs per month and owning exactly 2 vehicles. (ii) earning Rs or more per month and owning exactly 1 vehicle. (iii) earning less than Rs 7000 per month and does not own any vehicle. (iv) earning Rs per month and owning more than 2 vehicles. (v) owning not more than 1 vehicle. 6. Refer to Table 14.7, Chapter 14. (i) Find the probability that a student obtained less than 20% in the mathematics test. (ii) Find the probability that a student obtained marks 60 or above. 7. To know the opinion of the students about the subject statistics, a survey of 200 students was conducted. The data is recorded in the following table. Opinion Number of students like 135 dislike 65 Find the probability that a student chosen at random (i) likes statistics, (ii) does not like it. 8. Refer to Q.2, Exercise What is the empirical probability that an engineer lives: (i) less than 7 km from her place of work? (ii) more than or equal to 7 km from her place of work? (iii) within 1 km from her place of work? 2

15 PROBABILLITY Activity : Note the frequency of two-wheelers, three-wheelers and four-wheelers going past during a time interval, in front of your school gate. Find the probability that any one vehicle out of the total vehicles you have observed is a two-wheeler. 10. Activity : Ask all the students in your class to write a 3-digit number. Choose any student from the room at random. What is the probability that the number written by her/him is divisible by 3? Remember that a number is divisible by 3, if the sum of its digits is divisible by Eleven bags of wheat flour, each marked 5 kg, actually contained the following weights of flour (in kg): Find the probability that any of these bags chosen at random contains more than 5 kg of flour. 12. In Q.5, Exercise 14.2, you were asked to prepare a frequency distribution table, regarding the concentration of sulphur dioxide in the air in parts per million of a certain city for 30 days. Using this table, find the probability of the concentration of sulphur dioxide in the interval on any of these days. 13. In Q.1, Exercise 14.2, you were asked to prepare a frequency distribution table regarding the blood groups of 30 students of a class. Use this table to determine the probability that a student of this class, selected at random, has blood group AB Summary In this chapter, you have studied the following points: 1. An event for an experiment is the collection of some outcomes of the experiment. 2. The empirical (or experimental) probability P(E) of an event E is given by P(E) = Number of trials in which E has happened Total number of trials 3. The Probability of an event lies between 0 and 1 (0 and 1 inclusive).

### MAT 1000. Mathematics in Today's World

MAT 1000 Mathematics in Today's World We talked about Cryptography Last Time We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities

### Sample Space, Events, and PROBABILITY

Sample Space, Events, and PROBABILITY In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.

### 6.3 Probabilities with Large Numbers

6.3 Probabilities with Large Numbers In general, we can t perfectly predict any single outcome when there are numerous things that could happen. But, when we repeatedly observe many observations, we expect

### Section 6.2 Definition of Probability

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 that it will

### 33 Probability: Some Basic Terms

33 Probability: Some Basic Terms In this and the coming sections we discuss the fundamental concepts of probability at a level at which no previous exposure to the topic is assumed. Probability has been

### M2L1. Random Events and Probability Concept

M2L1 Random Events and Probability Concept 1. Introduction In this lecture, discussion on various basic properties of random variables and definitions of different terms used in probability theory and

### Mathematical goals. Starting points. Materials required. Time needed

Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about

### Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

### MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes

### An approach to Calculus of Probabilities through real situations

MaMaEuSch Management Mathematics for European Schools http://www.mathematik.unikl.de/ mamaeusch An approach to Calculus of Probabilities through real situations Paula Lagares Barreiro Federico Perea Rojas-Marcos

### Lesson 1: Experimental and Theoretical Probability

Lesson 1: Experimental and Theoretical Probability Probability is the study of randomness. For instance, weather is random. In probability, the goal is to determine the chances of certain events happening.

### Probability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes

Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all

### I. WHAT IS PROBABILITY?

C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and

### STOCHASTIC MODELING. Math3425 Spring 2012, HKUST. Kani Chen (Instructor)

STOCHASTIC MODELING Math3425 Spring 212, HKUST Kani Chen (Instructor) Chapters 1-2. Review of Probability Concepts Through Examples We review some basic concepts about probability space through examples,

### 36 Odds, Expected Value, and Conditional Probability

36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face

### An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event

An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event is the sum of the probabilities of the outcomes in the

### Probability Review. ICPSR Applied Bayesian Modeling

Probability Review ICPSR Applied Bayesian Modeling Random Variables Flip a coin. Will it be heads or tails? The outcome of a single event is random, or unpredictable What if we flip a coin 10 times? How

### Grade 7/8 Math Circles Fall 2012 Probability

1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Probability Probability is one of the most prominent uses of mathematics

### PROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA

PROBABILITY Thabisa Tikolo STATISTICS SOUTH AFRICA Probability is a topic that some educators tend to struggle with and thus avoid teaching it to learners. This is an indication that teachers are not yet

Chapter 3 Probability 2012 Pearson Education, Inc. All rights reserved. 1 of 20 Chapter Outline 3.1 Basic Concepts of Probability 3.2 Conditional Probability and the Multiplication Rule 3.3 The Addition

### 7.1 Sample space, events, probability

7.1 Sample space, events, probability In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.

### Lab 11. Simulations. The Concept

Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that

### PROBABILITY. Chapter Overview

Chapter 6 PROBABILITY 6. Overview Probability is defined as a quantitative measure of uncertainty a numerical value that conveys the strength of our belief in the occurrence of an event. The probability

### Probability, statistics and football Franka Miriam Bru ckler Paris, 2015.

Probability, statistics and football Franka Miriam Bru ckler Paris, 2015 Please read this before starting! Although each activity can be performed by one person only, it is suggested that you work in groups

### Rock, Paper, Scissors Tournament

UNIT 5: PROBABILITY Introduction to The Mathematics of Chance Rock, Paper, Scissors Tournament History As old as civilization. Egyptians used a small mammal bone as a 4 sided die (500 BC) Games of chance

### Probability definitions

Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a data-generating

### Concepts of Probability

Concepts of Probability Trial question: we are given a die. How can we determine the probability that any given throw results in a six? Try doing many tosses: Plot cumulative proportion of sixes Also look

### Solutions to Self-Help Exercises 1.3. b. E F = /0. d. G c = {(3,4),(4,3),(4,4)}

1.4 Basics of Probability 37 Solutions to Self-Help Exercises 1.3 1. Consider the outcomes as ordered pairs, with the number on the bottom of the red one the first number and the number on the bottom of

### Session 8 Probability

Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome

### Introduction to Probability

Chapter 4 Introduction to Probability 4.1 Introduction Probability is the language we use to model uncertainty. We all intuitively understand that few things in life are certain. There is usually an element

### Unit 19: Probability Models

Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,

### Most of us would probably believe they are the same, it would not make a difference. But, in fact, they are different. Let s see how.

PROBABILITY If someone told you the odds of an event A occurring are 3 to 5 and the probability of another event B occurring was 3/5, which do you think is a better bet? Most of us would probably believe

### 4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

### P (A) = lim P (A) = N(A)/N,

1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

### AP Stats - Probability Review

AP Stats - Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose

### Teaching & Learning Plans. Plan 2: Probability and Relative Frequency. Junior Certificate Syllabus Leaving Certificate Syllabus

Teaching & Learning Plans Plan 2: Probability and Relative Frequency Junior Certificate Syllabus Leaving Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what

### PROBABILITY 14.3. section. The Probability of an Event

4.3 Probability (4-3) 727 4.3 PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques

### Section 6-5 Sample Spaces and Probability

492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)

### PROBABILITY NOTIONS. Summary. 1. Random experiment

PROBABILITY NOTIONS Summary 1. Random experiment... 1 2. Sample space... 2 3. Event... 2 4. Probability calculation... 3 4.1. Fundamental sample space... 3 4.2. Calculation of probability... 3 4.3. Non

### Examples of infinite sample spaces. Math 425 Introduction to Probability Lecture 12. Example of coin tosses. Axiom 3 Strong form

Infinite Discrete Sample Spaces s of infinite sample spaces Math 425 Introduction to Probability Lecture 2 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 4,

### FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint

### What is the probability of throwing a fair die and receiving a six? Introduction to Probability. Basic Concepts

Basic Concepts Introduction to Probability A probability experiment is any experiment whose outcomes relies purely on chance (e.g. throwing a die). It has several possible outcomes, collectively called

### A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes.

Chapter 7 Probability 7.1 Experiments, Sample Spaces, and Events A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes. Each outcome

### Toss a coin twice. Let Y denote the number of heads.

! Let S be a discrete sample space with the set of elementary events denoted by E = {e i, i = 1, 2, 3 }. A random variable is a function Y(e i ) that assigns a real value to each elementary event, e i.

### Comparing relative frequency and probability

Comparing relative frequency and probability The mathematics of probability is concerned with things that happen at random, or unpredictably. A common example is a (fair) raffle in which each ticket has

### MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution

Chance deals with the concepts of randomness and the use of probability as a measure of how likely it is that particular events will occur. A National Statement on Mathematics for Australian Schools, 1991

### PROBABILITY. The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE

PROBABILITY 53 Chapter 3 PROBABILITY The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE 3. Introduction In earlier Classes, we have studied the probability as

### + Section 6.2 and 6.3

Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities

### Probability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2

Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability

### Consider a system that consists of a finite number of equivalent states. The chance that a given state will occur is given by the equation.

Probability and the Chi-Square Test written by J. D. Hendrix Learning Objectives Upon completing the exercise, each student should be able: to determine the chance that a given state will occur in a system

### REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.

REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game

### number of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.

12 Probability 12.1 Basic Concepts Start with some Definitions: Experiment: Any observation of measurement of a random phenomenon is an experiment. Outcomes: Any result of an experiment is called an outcome.

### Unit 18: Introduction to Probability

Unit 18: Introduction to Probability Summary of Video There are lots of times in everyday life when we want to predict something in the future. Rather than just guessing, probability is the mathematical

### THE MULTINOMIAL DISTRIBUTION. Throwing Dice and the Multinomial Distribution

THE MULTINOMIAL DISTRIBUTION Discrete distribution -- The Outcomes Are Discrete. A generalization of the binomial distribution from only 2 outcomes to k outcomes. Typical Multinomial Outcomes: red A area1

A Bit About the St. Petersburg Paradox Jesse Albert Garcia March 0, 03 Abstract The St. Petersburg Paradox is based on a simple coin flip game with an infinite expected winnings. The paradox arises by

### Possibilities and Probabilities

Possibilities and Probabilities Counting The Basic Principle of Counting: Suppose that two experiments are to be performed. Then if experiment 1 can result in any one of m possible outcomes and if, for

### 6th Grade Lesson Plan: Probably Probability

6th Grade Lesson Plan: Probably Probability Overview This series of lessons was designed to meet the needs of gifted children for extension beyond the standard curriculum with the greatest ease of use

### If a tennis player was selected at random from the group, find the probability that the player is

Basic Probability. The table below shows the number of left and right handed tennis players in a sample of 0 males and females. Left handed Right handed Total Male 3 29 32 Female 2 6 8 Total 4 0 If a tennis

### Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability

Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock

### EDEXCEL FUNCTIONAL SKILLS PILOT

EEXEL FUNTIONAL SKILLS PILOT Maths Level hapter 7 Working with probability SETION K Measuring probability 9 Remember what you have learned 3 raft for Pilot Functional Maths Level hapter 7 Pearson Education

### Chapter 4: Probability and Counting Rules

Chapter 4: Probability and Counting Rules Learning Objectives Upon successful completion of Chapter 4, you will be able to: Determine sample spaces and find the probability of an event using classical

### Slide 1 Math 1520, Lecture 23. This lecture covers mean, median, mode, odds, and expected value.

Slide 1 Math 1520, Lecture 23 This lecture covers mean, median, mode, odds, and expected value. Slide 2 Mean, Median and Mode Mean, Median and mode are 3 concepts used to get a sense of the central tendencies

### Definition and Calculus of Probability

In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the

### Grade 7 Mathematics. Unit 7. Data Analysis. Estimated Time: 18 Hours

Grade 7 Mathematics Data Analysis Estimated Time: 18 Hours [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] Problem Solving [R] Reasoning [T] Technology [V] Visualization

### Solutions: Problems for Chapter 3. Solutions: Problems for Chapter 3

Problem A: You are dealt five cards from a standard deck. Are you more likely to be dealt two pairs or three of a kind? experiment: choose 5 cards at random from a standard deck Ω = {5-combinations of

### Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.

1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event

### Chapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.

MATH 11008: Probability Chapter 15 Definitions: experiment: is the act of making an observation or taking a measurement. outcome: one of the possible things that can occur as a result of an experiment.

### That s Not Fair! ASSESSMENT #HSMA20. Benchmark Grades: 9-12

That s Not Fair! ASSESSMENT # Benchmark Grades: 9-12 Summary: Students consider the difference between fair and unfair games, using probability to analyze games. The probability will be used to find ways

### Chapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.

Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive

### Probability (Day 1 and 2) Blue Problems. Independent Events

Probability (Day 1 and ) Blue Problems Independent Events 1. There are blue chips and yellow chips in a bag. One chip is drawn from the bag. The chip is placed back into the bag. A second chips is then

### Chapter 4 Probability

The Big Picture of Statistics Chapter 4 Probability Section 4-2: Fundamentals Section 4-3: Addition Rule Sections 4-4, 4-5: Multiplication Rule Section 4-7: Counting (next time) 2 What is probability?

### AMS 5 CHANCE VARIABILITY

AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and

### People have thought about, and defined, probability in different ways. important to note the consequences of the definition:

PROBABILITY AND LIKELIHOOD, A BRIEF INTRODUCTION IN SUPPORT OF A COURSE ON MOLECULAR EVOLUTION (BIOL 3046) Probability The subject of PROBABILITY is a branch of mathematics dedicated to building models

### Ch. 13.3: More about Probability

Ch. 13.3: More about Probability Complementary Probabilities Given any event, E, of some sample space, U, of a random experiment, we can always talk about the complement, E, of that event: this is the

### The number of phone calls to the attendance office of a high school on any given school day A) continuous B) discrete

Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) State whether the variable is discrete or continuous.

### 1. The sample space S is the set of all possible outcomes. 2. An event is a set of one or more outcomes for an experiment. It is a sub set of S.

1 Probability Theory 1.1 Experiment, Outcomes, Sample Space Example 1 n psychologist examined the response of people standing in line at a copying machines. Student volunteers approached the person first

### Mental Questions. Day 1. 1. What number is five cubed? 2. A circle has radius r. What is the formula for the area of the circle?

Mental Questions 1. What number is five cubed? KS3 MATHEMATICS 10 4 10 Level 8 Questions Day 1 2. A circle has radius r. What is the formula for the area of the circle? 3. Jenny and Mark share some money

### Chapter 16: law of averages

Chapter 16: law of averages Context................................................................... 2 Law of averages 3 Coin tossing experiment......................................................

### Distributions. and Probability. Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment

C Probability and Probability Distributions APPENDIX C.1 Probability A1 C.1 Probability Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment When assigning

### Lesson 8: The Difference Between Theoretical Probabilities and Estimated Probabilities

The Difference Between Theoretical Probabilities and Estimated Probabilities Student Outcomes Given theoretical probabilities based on a chance experiment, students describe what they expect to see when

### Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Lecture 16: More Box Models Tessa L. Childers-Day UC Berkeley 22 July 2014 By the end of this lecture... You will be able to: Determine what we expect the sum

### Law of Large Numbers. Alexandra Barbato and Craig O Connell. Honors 391A Mathematical Gems Jenia Tevelev

Law of Large Numbers Alexandra Barbato and Craig O Connell Honors 391A Mathematical Gems Jenia Tevelev Jacob Bernoulli Life of Jacob Bernoulli Born into a family of important citizens in Basel, Switzerland

### Basic concepts in probability. Sue Gordon

Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are

### Coin Tossing: Teacher-Led Lesson Plan

: Teacher-Led Lesson Plan Subject/Strand/Topic: Math Data Management & Probability Grade(s) / Course(s): 7 Ontario Expectations: 7m85, 7m86 Key Concepts: probability of tossing a coin Link: http://nlvm.usu.edu/en/nav/frames_asid_305_g_3_t_5.html

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

Ch. - Problems to look at 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

### Expected Value and the Game of Craps

Expected Value and the Game of Craps Blake Thornton Craps is a gambling game found in most casinos based on rolling two six sided dice. Most players who walk into a casino and try to play craps for the

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Exam Name 1) Solve the system of linear equations: 2x + 2y = 1 3x - y = 6 2) Consider the following system of linear inequalities. 5x + y 0 5x + 9y 180 x + y 5 x 0, y 0 1) 2) (a) Graph the feasible set

### Chapter 4. Probability and Probability Distributions

Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the

### Chapter 13 & 14 - Probability PART

Chapter 13 & 14 - Probability PART IV : PROBABILITY Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14 - Probability 1 / 91 Why Should We Learn Probability Theory? Dr. Joseph

### CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction

CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous

### 4.4 Conditional Probability

4.4 Conditional Probability It is often necessary to know the probability of an event under restricted conditions. Recall the results of a survey of 100 Grade 12 mathematics students in a local high school.

### Mathematics in the New Zealand Curriculum Second Tier

Mathematics in the New Zealand Curriculum Second ier Strand: Statistics hread: Probability Level: Four Achievement Objective: Investigate situations that involve elements of chance by comparing experimental

### Jan 17 Homework Solutions Math 151, Winter 2012. Chapter 2 Problems (pages 50-54)

Jan 17 Homework Solutions Math 11, Winter 01 Chapter Problems (pages 0- Problem In an experiment, a die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample

### Introduction and Overview

Introduction and Overview Probability and Statistics is a topic that is quickly growing, has become a major part of our educational program, and has a substantial role in the NCTM Standards. While covering

### Revision Notes Adult Numeracy Level 1

Revision Notes Adult Numeracy Level 1 Numbers The number 5 703 428 has been entered into a table. It shows the value of each column. The 7 is in the hundred thousands column The 0 cannot be missed out

### JANUARY TERM 2012 FUN and GAMES WITH DISCRETE MATHEMATICS Module #9 (Geometric Series and Probability)

JANUARY TERM 2012 FUN and GAMES WITH DISCRETE MATHEMATICS Module #9 (Geometric Series and Probability) Author: Daniel Cooney Reviewer: Rachel Zax Last modified: January 4, 2012 Reading from Meyer, Mathematics