Mathematics
Probability Rules
Probability rules are a set of principles used to calculate the likelihood of events occurring in a given situation. These rules include the addition rule, multiplication rule, complement rule, and conditional probability. The addition rule deals with the probability of either of two events happening, while the multiplication rule calculates the probability of both events occurring. The complement rule finds the probability of an event not happening, and conditional probability assesses the likelihood of an event given that another event has occurred.
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No longer available |Learn moreUnderstandable Statistics
Concepts and Methods, Enhanced
- Charles Henry Brase, Corrinne Pellillo Brase(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Other important ways will be discussed later. • The law of large numbers indicates that as the number of trials of a statistical experiment or observation increases, the relative frequency of a designated event becomes closer to the theoretical probability of that event. • Events are mutually exclusive if they cannot occur together. Events are independent if the occurrence of one event does not change the probability of the occurrence of the other. • Conditional probability is the probability that one event will occur, given that another event has occurred. • The complement rule gives the probability that an event will not occur. The addition rule gives the probability that at least one of two specified events will occur. The multiplication rule gives the probability that two events will occur together. • To determine the probability of equally likely events, we need to know how many outcomes are possible. Devices such as tree diagrams and counting rules—such as the multiplication rule of counting, the permutations rule, and the combinations rule—help us determine the total number of outcomes of a statistical experiment or observation. In most of the statistical applications of later chapters, we will use the addition rule for mutu-ally exclusive events and the multiplication rule for independent events. Copyright 201 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
- Teresa Bradley(Author)
- 2014(Publication Date)
- Wiley(Publisher)
P R O B A B I L I T Y 4 4.1 Introduction to probability 4.2 Multiplication and addition rules for probability 4.3 Joint, marginal and conditional probability 4.4 Bayes’ Rule Chapter Objectives Having carefully studied this chapter and completed the exercises you should be able to do the following Define and describe probability in terms of relative frequency and equally likely events Explain the axioms (self evident truths) of probability State and apply the counting rules for probability Define and describe independent events, mutually exclusive events and dependent events State the ‘AND’ and ‘OR’ rules Apply and solve problems based on the ‘AND’ and ‘OR’ rules Solve problem involving conditional probability Define a probability distribution Define and calculate joint, marginal and conditional probability distributions from a contingency table Derive and explain Bayes’ Theorem Solve problems based on Bayes’ Theorem 4.1 Introduction to probability Probability is essentially a measure of the likelihood or chance of ‘something’ happening. Some examples to illustrate the use of probability in everyday language are given below: 1. ‘There is a 50:50 chance that a head will turn up when you toss a coin’; 2. ‘The probability of getting an even number on one toss of a die is 0.5’; [ 160 ] C H A P T E R 4 3. ‘The odds of passing this exam are 3 to 1’; 4. ‘The odds of that horse winning are 100: 1’; 5. ‘There is a 30 % chance of a child developing mumps when more than two children in the class are already infected’; 6. ‘There is 1 chance in 3 of getting a “1 or a 6” on one roll of a die’; 7. ‘There is a 0.001% chance of an allergic reaction to that medication’; 8. ‘There is 1 chance in 10 000 that the new PC will breakdown in the first 2 years’, etc., etc.
eBook - ePub- Dennis V. Lindley(Author)
- 2013(Publication Date)
- Wiley(Publisher)
With this terminology, the last form of the multiplication rule reads that your probability of the product of two independent events is the product of their separate probabilities, a result that is attractive because it is easy to remember. Unfortunately, it is true only if the events are independent; otherwise it is wrong, and often seriously wrong. Similarly, (5.2) reads that your probability of the sum of two events is the sum of their separate probabilities. Again, this is true only under restrictions, but this time the restriction is not independence but the requirement that the events be exclusive. Simple as these special forms are, their simplicity can easily lead to errors and are therefore best avoided unless the restrictions that made them valid are always remembered throughout the calculations. The desire for simplicity has often been emphasized, but here is an example where it is possible to go too far and think of the addition and multiplication rules in their simpler forms, forgetting the restrictions that must hold before they are correct. Notice that the restriction, necessary for the simple form of the addition rule, that the events be exclusive, or the disjunction impossible, is a logical restriction, having nothing to do with uncertainty, whereas independence, the restriction with the multiplication rule, is essentially probabilistic. It is perhaps pedantic to point out that the simple form of the addition rule is correct if you judge the disjunction to have probability zero, rather than knowing it is logically impossible, but we will see in §6.8 that it is dangerous to attach probability zero to anything other than a logical impossibility. 5.4 The Basic Rules There are now two rules that your probabilities have to obey: addition and multiplication
- Brase/Brase, Charles Henry Brase, Corrinne Pellillo Brase(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
A standard notation for P ( A, given B ) is P 1 A | B 2 . GENERAL MULTIPLICATION RULE FOR ANY EVENTS P 1 A and B 2 5 P 1 A 2 # P 1 B | A 2 (5) P 1 A and B 2 5 P 1 B 2 # P 1 A | B 2 (6) We will use either formula (5) or formula (6) according to the information available. Formulas (4), (5), and (6) constitute the multiplication rules of probability . They help us compute the probability of events happening together when the sample space is too large for convenient reference or when it is not completely known. Note: For conditional probability, observe that the multiplication rule P 1 A and B 2 5 P 1 B 2 # P 1 A | B 2 can be solved for P 1 A | B 2 , leading to CONDITIONAL PROBABILITY (WHEN P ( B ) ? 0) P 1 A | B 2 5 P 1 A and B 2 P 1 B 2 Probability of A and B Dependent events Conditional probability P 1 A | B 2 Multiplication rules of probability LOOKING FORWARD Important techniques for using conditional prob-abilities were discovered by Reverend Thomas Bayes (1702–1761), an English mathematician. You can find a discussion regarding Bayes´s theorem in the appendix of Understandable Statistics 11th edition by Brase and Brase. This content is also available through custom pub-lishing with Understanding Basic Statistics . Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 5.2 Some Probability Rules—Compound Events 203 We will see some applications of this formula in later chapters. Let’s use the multiplication rules to complete the dice and ball-drawing prob-lems just discussed. We’ll compare the results with those obtained by using the sam-ple space directly.
eBook - PDFStatistics
Learning from Data
- Roxy Peck, Tom Short(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
In other situations, you started with prob-ability information and used that information to construct a hypothetical 1000 table that could be used to calculate probabilities of interest. In this section, you will see that it is possible to bypass the construction of a hypothetical 1000 table and use a few key prob-ability formulas to calculate probabilities directly. Probability Formulas The Complement Rule (for calculating complement probabilities) For any event E , P ( E C ) 5 1 2 P ( E ) The Addition Rule (for calculating union probabilities) For any two events E and F , P ( E ∪ F ) 5 P ( E ) 1 P ( F ) 2 P ( E ∩ F ) For mutually exclusive events , this simplifies to P ( E ∪ F ) 5 P ( E ) 1 P ( F ) The Multiplication Rule (for calculating intersection probabilities) For any two events E and F , P ( E ∩ F ) 5 P ( F ) P ( E|F ) 5 P ( E|F ) P ( F ) For independent events, this simplifies to P ( E ∩ F ) 5 P ( E ) P ( F ) Conditional Probabilities For any two events E and F , with P ( F ) ≠ 0, P ( E | F ) 5 P ( E ∩ F ) ________ P ( F ) Three of the probability formulas in the preceding box were introduced in Section 5.3 (the Complement Rule, the Addition Rule for mutually exclusive events, and the Multiplication Rule for independent events). In the following examples, you will see how the other formulas follow from the work you have already done with hypothetical 1000 tables. Union Probabilities: The Addition Rule To see how the Addition Rule can be used to calculate the probability of a union event, let’s revisit the health information study first introduced in Example 5.5. Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.
- William DeCoursey(Author)
- 2003(Publication Date)
- Newnes(Publisher)
In this chapter we examine the basic ideas and approaches to probability and its calculation. We look at calculating the probabilities of combined events. Under some circumstances probabilities can be found by using counting theory involving permu-tations and combinations. The same ideas can be applied to somewhat more complex situations, some of which will be examined in this chapter. 2.1 Fundamental Concepts (a) Probability as a specific term is a measure of the likelihood that a particular event will occur. Just how likely is it that the outcome of a trial will meet a particular requirement? If we are certain that an event will occur, its probability is 1 or 100%. If it certainly will not occur, its probability is zero. The first situation corresponds to an event which occurs in every trial, whereas the second corresponds to an event which never occurs. At this point we might be tempted to say that probability is given by relative frequency, the fraction of all the trials in a particular experiment that give an outcome meeting the stated requirements. But in general that would not be right. Why? Because the outcome of each trial is determined by chance. Say we toss a fair coin, one which is just as likely to give heads as tails. It is entirely possible that six tosses of the coin would give six heads or six tails, or anything in between, so the relative frequency of heads would vary from zero to one. If it is just as likely that an event will occur as that it will not occur, its true probability is 0.5 or 50%. But the experiment might well result in relative frequencies all the way from zero to one. Then the relative frequency from a small number of trials gives a very unreliable indication of probability. In section 5.3 we will see how to make more quantitative calcula-tions concerning the probabilities of various outcomes when coins are tossed randomly or similar trials are made.
eBook - PDFStatistics Using R
An Integrative Approach
- Sharon Lawner Weinberg, Daphna Harel, Sarah Knapp Abramowitz(Authors)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
Keep in mind that only within the context of a well-defined experiment are events and their probabilities of occurrence defined. + Remark. Notice that the probability of an event is its relative frequency. As such, probability values are often reported as percentages. THE COMPLEMENT RULE OF PROBABILITY Let us take another look at parts (1) and (2) of Example 7.4. Because males and females have no possible outcomes in common (in other words, a female individual cannot also be male and vice versa), we call these two events mutually exclusive or disjoint. In addition, because there are only two categories, the event “The individual selected is male” may also be described as “The individual selected is not female. ” Accordingly, we say that the event “The individual selected is male” is the complement of the event “The individual selected is female.” Complement rule of probability: If two events E 1 and E 2 are complements of each other, then PðE 1 Þ ¼ 1–PðE 2 Þ: (7.2) Given that the probability is 0.546 that the individual selected is a female, we could have obtained the answer to part (2) of Example 7.4 as 1 – 0.546 = 0.454. + Remark. We often denote the complement of E 1 as E c 1 . The superscript c denotes that we are interested in the complementary event (to be described in more detail in a later section of this chapter). THE COMPLEMENT RULE OF PROBABILITY 223 THE ADDITIVE RULES OF PROBABILITY In part (4) of Example 7.4, we sought the probability of occurrence for the event: “The individual selected is either male or female,” which, in this case, equals 1.00 because all individuals in our sample are either male or female. We can approach the solution to this problem in a slightly different way.
eBook - PDF- L. Z. Rumshiskii(Author)
- 2016(Publication Date)
- Pergamon(Publisher)
(1.3) E V E N T S A N D P R O B A B I L I T I E S 5 (3) The probabilities of events obey the rule of addition : if the event C occurs when one of two mutually exclusive events A, Β occurs, then the probability of the event C is equal to the sum of the probabilities of the events A and B. We express this rule in the form: P{A or B) = F{A} + P{B} (for mutually exclusive A and B). (1.4) This relation is also known as the additivity property for prob-abilities. The first two conditions follow immediately from (1.1), because in this case Μ ^ 0, Ν > 0 and for the certain event Μ — Ν (all outcomes of a trial are favourable to the certain event). The third condition is verified in the following way for the scheme of random sampling. Let us suppose that there are Ν balls in an urn, of which Κ are red, L are blue, and the remainder are white ; a trial will consist of drawing one ball at random from the urn; the event A will occur if a red ball is drawn, and the event Β will occur if a blue ball is drawn. Then the event {A or B) consists in the drawing of a coloured ball (either red or blue). The direct calculation of the probabilities acording to (1.1) gives: ρμ} = —; P{2>} = — ; Ν Ν F{A or B} Κ + L Ν which agrees with (1.4). In order to apply the theory of probability it is extremely im-portant that these properties should be valid not only for the scheme of random sampling but also for any system of events. We can base such an assertion on the following argument. We recall that the general idea of probability arose from the observed stabil-ity of the relative frequency of an event. It is natural therefore to require that the basic properties of probabilities should coincide with the corresponding properties of relative frequencies. Now properties 1, 2 and 3 are easily verified for relative frequen-cies. ΕΡΤ. 2 6 ELEMENTS OF P R O B A B I L I T Y THEORY (l)f A relative frequency m/n cannot be negative since m ^ 0, η > 0.
eBook - PDF- Barbara Illowsky, Susan Dean(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
3 | PROBABILITY TOPICS Figure 3.1 Meteor showers are rare, but the probability of them occurring can be calculated. (credit: Navicore/flickr) Introduction Chapter Objectives By the end of this chapter, the student should be able to: • Understand and use the terminology of probability. • Determine whether two events are mutually exclusive and whether two events are independent. • Calculate probabilities using the Addition Rules and Multiplication Rules. • Construct and interpret Contingency Tables. • Construct and interpret Venn Diagrams. • Construct and interpret Tree Diagrams. It is often necessary to "guess" about the outcome of an event in order to make a decision. Politicians study polls to guess their likelihood of winning an election. Teachers choose a particular course of study based on what they think students can comprehend. Doctors choose the treatments needed for various diseases based on their assessment of likely results. You may have visited a casino where people play games chosen because of the belief that the likelihood of winning is good. You may have chosen your course of study based on the probable availability of jobs. You have, more than likely, used probability. In fact, you probably have an intuitive sense of probability. Probability deals with the chance of an event occurring. Whenever you weigh the odds of whether or not to do your homework or to study for an exam, you are using probability. In this chapter, you will learn how to solve probability problems using a systematic approach. Your instructor will survey your class. Count the number of students in the class today. Chapter 3 | Probability Topics 171 • Raise your hand if you have any change in your pocket or purse. Record the number of raised hands. • 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.
eBook - PDF- Prem S. Mann(Author)
- 2016(Publication Date)
- Wiley(Publisher)
129 129 CHAPTER 4 We often make statements about probability. For example, a weather forecaster may predict that there is an 80% chance of rain tomorrow. A study may predict that a female, compared to a male, has a higher probability of being in favor of gun control. A college student may ask an instructor about the chances of passing a course or getting an A if he or she did not do well on the midterm examination. Probability, which measures the likelihood that an event will occur, is an important part of statis- tics. It is the basis of inferential statistics, which will be introduced in later chapters. In inferential statistics, we make decisions under conditions of uncertainty. Probability theory is used to evaluate the uncertainty involved in those decisions. For example, estimating next year’s sales for a company is based on many assumptions, some of which may happen to be true and others may not. Probabil- ity theory will help us make decisions under such conditions of imperfect information and uncertainty. Combining probability and probability distributions (which are discussed in Chapters 5 through 7) with descriptive statistics will help us make decisions about populations based on information obtained from samples. This chapter presents the basic concepts of probability and the rules for computing probability. Do you worry about your weight? According to a Gallup poll, 45% of American adults worry all or some of the time about their weight. The poll showed that more women than men worry about their weight all or some of the time. In this poll, 55% of adult women and 35% of adult men said that they worry all or some of the time about their weight.
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