Mathematics
Probability of Combined Events
The probability of combined events in mathematics refers to the likelihood of two or more events occurring together. It is calculated using the principles of probability, such as the multiplication rule for independent events or the addition rule for mutually exclusive events. Understanding the probability of combined events is essential for making informed decisions in various real-world scenarios.
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11 Key excerpts on "Probability of Combined Events"
- 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 - PDF- Kathleen Subrahmaniam(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
. . , k determine the probability model for a random experiment. From the probability model we can obtain the probability of any event asso-ciated with the experiment. DEFINITION 2.4 The probability of any event E is the sum of the prob-abilities of the simple events which constitute the event E. Two somewhat special cases arise: the entire sample space and the impossible event. Since all the possible outcomes of an experiment must be enumerated in the sample space, Pr(S) = 1. This would be translated as som e event in S must occu r,n which seems very reasonable from the defi-nition of S. If any event is not a possible outcome of the experiment, then it has no corresponding sample points in S. We will call this event an im pos-sible event and its probability is obviously zero. 12 Chapter 2 Referring to our penny-nickel experiment, let us develop an appropri-ate probability model and calculate the probabilities corresponding to the events U, V, W, X, Y and Z. It would seem reasonable, and it can be veri-fied by experimentation, that each outcome is equally likely. Assigning probability 1/4 to each point in S 2 and using Definition 2.4, we find that Pr(U) = 1/2, Pr(V) = 1/4, Pr(W) = 3/4, Pr(X) = 1/2, Pr(Y) = 1/2 and Pr(Z) = 1/4. 2.3 COMBINING EVENTS Since sets and events are analogous, we will now discuss how events, like sets, may be combined. In combining events we are faced with the problem of translating words into logical expressions» In everyday usage, expres-sions of the form A or B” may be interpreted in two different ways: 1. Exclusive A or B but not both 2. Inclusive A or B or both In the following discussion we shall restrict ourselves to the inclusive form . Simultaneous membership in two sets A and B is expressed in words by the terminology and. DEFINITION 2.6 The intersection of the events A and B in S is the set of all points belonging to A and to B» A and B = AB = A fl B = {x |x e A and x < e B }.
eBook - PDF- Prem S. Mann(Author)
- 2016(Publication Date)
- Wiley(Publisher)
It is obvious from the formula for joint probability that if we know the probability of an event A and the joint probability of events A and B, then we can calculate the conditional probability of B given A. Calculating Conditional Probability If A and B are two events, then, P( B ∣ A) = P( A and B) P( A) and P( A ∣ B) = P( A and B) P( B) given that P( A) ≠ 0 and P( B) ≠ 0. EXAMPLE 4–24 Major and Status of a Student The probability that a randomly selected student from a college is a senior is .20, and the joint probability that the student is a computer science major and a senior is .03. Find the conditional probability that a student selected at random is a computer science major given that the student is a senior. Solution From the given information, the probability that a randomly selected student is a senior is: P( senior ) = .20 The probability that the selected student is a senior and computer science major is: P( senior and computer science major ) = .03 Hence, the conditional probability that a randomly selected student is a computer science major given that he or she is a senior is: P( computer science major ∣ senior ) = P( senior and computer science major ) ∕ P( senior ) = .03 .20 = .15 Thus, the (conditional) probability is .15 that a student selected at random is a computer science major given that he or she is a senior. ◼ Calculating the conditional probability of an event. 4.4.4 Joint Probability of Mutually Exclusive Events We know from an earlier discussion that two mutually exclusive events cannot happen together. Consequently, their joint probability is zero. 4.4 Intersection of Events and the Multiplication Rule 155 EXAMPLE 4–25 Car Loan Application Consider the following two events for an application filed by a person to obtain a car loan: A = event that the loan application is approved R = event that the loan application is rejected What is the joint probability of A and R? Solution The two events A and R are mutually exclusive.
eBook - PDF- L. Z. Rumshiskii(Author)
- 2016(Publication Date)
- Pergamon(Publisher)
We remark that by the classical definition the probability of an event is equal to zero if and only if there is no possible outcome of a trial which is favourable to this event (i.e. M = 0 ) . We shall see later in the study of continuous random variables that an event whose probability is zero is not necessarily impossible. The relation between the classical definition and the fundamental properties As we emphasised above, the classical definition of probability is concerned with the case where the outcomes of a trial can be represented as a partition into equally likely events. It is worth noticing that in this case the classical for-mula (1.1) can be deduced from the three fundamental properties of proba-bilities. We prove the following assertion : If the elementary outcomes of a trial form a partition of equally likely events, then the probability of a compound event is given by the classical formula (1.1). To prove this we denote the elementary outcomes by E t , E 2 ,..., E N and we write their common probability as ρ = P{E k } (k = 1,2,..., N). From (1.6) we have P{E i } + F{E 2 } + — + PiE N } = l 9 so that Np = 1. A is a compound event for which M given elementary outcomes (Ει, E 2 ,..., E M , say) are favourable. Then A is the compound event (£Ί or E 2 or ... or E M ) and applying (1.5) we obtain PM} = P[E l } + P{E 2 ] + ·'· + P[E M ] so that V{A} = Mp = — . Ν § 4. THE I N T E R S E C T I O N OF EVENTS. I N D E P E N D E N T E V E N T S The event consisting of the simultaneous occurrence of the events A and Β is called the intersection of the events A and B. We shall denote this event by (A and B). EXAMPLE . We pick one number at random from the numbers 1, 2 , 1 0 0 . The event A consists of the numbers divisible by 3 and the event Β consists of the numbers divisible by 4 . Then the event {A and B) consists of the numbers divisible by both 3 and 4 , E V E N T S A N D P R O B A B I L I T I E S 9 i.e. the numbers divisible by 12.
eBook - PDF- D.G. Rees(Author)
- 2018(Publication Date)
- Chapman and Hall/CRC(Publisher)
We all use ‘subjective probability’ in forecasting future events, for example, when we try to decide whether it will rain tomorrow, and when we try to assess the reactions of others to our opinions and actions. We may not be quite so calculating as to estimate a probability value, but we may regard future events as being probable, rather than just possible. In subjective assessments of probability we may take into account experimental data from past events, but we are likely to add a dose of subjectivity depending on our personality, our mood, and other factors. 5.8 Probabilities Involving More Than One Event Suppose that we are interested in the probabilities of two possible events, El and E 2. For example, we may wish to know the probability that both events will occur, or perhaps the probability that either or both events will occur. We will refer to these as, respectively, P(El and E 2) and P{E or E or both). In set theory notation these compound events are called the intersec-tion and union of events E and E 2, and their probabilities are written: P(El PI E2) and P(E1 U E2) Probability ■ 53 There are two probability laws which can be used to estimate such probabilities, and these are discussed in Sections 5.9 and 5.10. 5.9 Multiplication Law (The 'and' Law) The general case of the m ultiplication law is PiE, and E 2) = P iE JP iE JE J (5.3) where P(E E1) means the probability that event E will occur, given that event E has already occurred. The vertical line between E and E should be read as ‘given that’ or ‘on the condition that’. P(E El ) is an example of what is called a conditional probability. Probability Example 5.4 If two cards are selected at random, one at a time w ithout replacem ent from a pack of 52 playing cards, what is the probability that both cards will be aces? P(two aces) = P(first card is ace and second card is ace), which is logical.
eBook - PDFMathematics NQF4 SB
TVET FIRST
- M Van Rensburg, I Mapaling M Trollope(Authors)
- 2017(Publication Date)
- Macmillan(Publisher)
354 Module 13 Use experiments, simulations and probability distribution to set and explore probability models Module 13 Overview By the end of this module you should be able to: • Unit 13.1: Explain and distinguish between the following terminology/events: – Probability. – Dependent events. – Independent events. – Mutually exclusive events. – Mutually inclusive events. – Complementary events. • Unit 13.2: Make predictions based on validated experimental probabilities taking the following into account: – P(S) = 1 (where S is the sample space). – Disjoint (mutually exclusive) events P(A or B) = P(A) + P(B). – Complementary events, therefore being able to calculate the probability of an event not occurring. – P(A) or (B) = P(A) + P(B) − P(A and B) (where A and B are events within a sample space). – Correctly identify dependent and independent events and apply the product rule for independent events: P(A and B) = P(A) ∙ P(B). • Unit 13.3: Draw tree diagrams and Venn diagrams, complete two-way contingency tables to solve probability problems, and interpret and clearly communicate the results of experiments correctly in terms of real context. Unit 13.1: Probability terminology Probability is a measure of the relative likelihood of an event taking place. Or, put differently, it tells us how likely something is to happen. The extent to which something is probable can be found either by theoretical means or by doing an experiment. When we use theory to solve a probability problem, we use logical thinking and can apply a formula to solve the problem. When we conduct an experiment, we perform activities. Examples are when you toss a coin, roll a die or a pair of dice, select a card from a deck of cards, or a ball or marble from a bag of the relevant items. Table 13.1 explains the different probability terminology. Where applicable, the example shows how the term would be represented in a Venn diagram.
eBook - PDF- Ken Black, Ignacio Castillo, Amy Goldlist, Timothy Edmunds(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Combining Probabilities 141 This approach (calculating the probability of each indivisible part of the combined event and summing them together) is perfectly acceptable, but there are also other ways to express the formula for calculating the probability of the union of two events. In our final calculation of the probability P N S ( or ) , let’s replace the numbers we used with the formulas that they came from: P N S P N S ( ) ( or ) 0.14 0.56 0.11 ∪ = = + + P N P N S P N S P S P N S ( ( ) ( and )) ( and ) ( ( ) ( and )) = − + + − Simplifying this equation (by letting one of the negative P(N and S) cancel out the positive P(N and S), we are left with P N S P N P S P N S ( or ) ( ) ( ) ( and ) = + − This formula is known as the general law of addition. General Law of Addition P X Y P X P Y P X Y ( or ) ( ) ( ) ( and ) = + − P X Y P X P Y P X Y ( ) ( ) ( ) ( ) ∪ = + − ∩ where X, Y are events and X Y P X Y ( ) ( and ) ∩ = is the intersection of X and Y. (5.7) For example, let’s say you are considering two events: restaurant A going out of business in the next year (event A), and restaurant B going out of business in the next year (event B). Suppose that you know (somehow) that the probability of event A occurring is 0.5, the prob- ability of event B occurring is 0.3, and the probability of both A and B occurring is 0.1. You would like to know the probability of A or B. We can simply input these probabilities into the general law of addition to see that P A B P A P B P A B ( or ) ( ) ( ) ( and ) = + − 0.5 0.3 0.1 0.7 = + − = The Special Law of Addition If two events are mutually exclusive, the probability of the union of the two events is just the probability of the first event plus the probability of the sec- ond event. Because mutually exclusive events do not intersect, nothing has to be subtracted.
eBook - PDF- Howard G Tucker(Author)
- 1998(Publication Date)
- World Scientific(Publisher)
Chapter 1 Events and Probability 1.1 Introduction to Probability The notion of the probability of an event may be approached by at least three methods. One method, perhaps the first historically, is to repeat an experiment or game (in which a certain event might or might not occur) many times under identical conditions and compute the relative frequency with which the event occurs. This means: divide the total number of times that the specific event occurs by the total number of times the experiment is performed or the game is played. This ratio is called the relative frequency and is really only an approximation of what would be considered as the probability of the event. For example, if one tosses a penny 25 times, and if it comes up heads exactly 13 times, then we would estimate the probability that this particular coin will come up heads when tossed is 13/25 or 0.52. Although this method of arriving at the notion of probability is the most primitive and un-sophisticated, it is the most meaningful to the practical individual, in particular, to the working scientist and engineer who have to apply the results of probability theory to real-life situations. Accordingly, what-ever results one obtains in the theory of probability and statistics, one should be able to interpret them in terms of relative frequency. A sec-ond approach to the notion of probability is from an axiomatic point of view. That is, a minimal list of axioms is set down which assumes cer-tain properties of probabilities. From this minimal set of assumptions 1 2 CHAPTER 1. EVENTS AND PROBABILITY the further properties of probabiUty are deduced and applied. A third approach to the notion of probability is limited in applica-tion but is sufficient for our study of sample surveys. This approach is that of probabiUty in the equaUy likely case. Let us consider some game or experiment which, when played or performed, has among its possible outcomes a certain event E.
eBook - PDF- Jingmei Jiang(Author)
- 2022(Publication Date)
- Wiley(Publisher)
It equals the probability that both A and B occur divided by the probability that A occurs: P B A P A B P A P A | ( ) = ( ) ( ) ( ) > ∩ , assuming 0 (3.3) Definition 3.2 indicates that the probability of event B is related to whether event A occurs. Therefore, whether there is a correlation between two events can be studied through conditional probability. 3.3.2 Independence of Events Definition 3.3 For any events A and B , if P B A P B | ( ) = ( ) or P A ( ) = 0, then events A and B are called independent . Thus, Formula 3.3 can be written as P A B P A P B ∩ ( ) = ( ) × ( ) (3.4) Example 3.8 Given that the proportion of men in the Chinese population is 51.1% and the proportion of blood type AB is 7.0%, estimate the proportion of men with blood type AB in the Chinese population. 3.4 Multiplication Law of Probability 61 Solution Suppose that event A represents “man” and event B represents “blood type AB .” Then P A ( ) = 0 511 . and P B ( ) = 0 070 . . Because the ABO blood group system is not inherited in a sex-linked manner, the two events are independent of each other. Hence, based on Definition 3.3, the probability of men with blood type AB in the Chinese population is P A B P A P B ∩ ( ) = ( ) × ( ) = × = 0 511 0 070 0 036 . . . Thus, the estimated proportion of men with blood type AB in the Chinese population is 3.6%. From this example, we can see that, in biomedical research, the independence of events can be assessed based on experimental conditions and biological knowledge. The concept of “independence” can also be extended to n events. For events A A A n 1 2 , , , … , we say that the n events are independent if P A A A P A P A P A i i i i i i m m 1 2 1 2 ∩ ∩∩ ( ) = ( ) × ( ) × × ( ) for any subset of these n events i i i n m n m 1 2 1 2 , , , , , , … … ∈ { } ≤ ( ) , . Clearly, if the events in a group are independent, then any pairs of events are also independent.
No longer available |Learn more- William Mendenhall, Robert Beaver, Barbara Beaver, , William Mendenhall, Robert Beaver, Barbara Beaver(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
Reviewing the Probability Rules 1. The Addition Rule: The probability of a union of two events— P A B ( or or both) — can be calculated as 1 2 ( ) ( ) ( ) ( ) ∪ ∩ P A B P A P B P A B If A and B are mutually exclusive, then 1 ( ) ( ) ( ) ∪ P A B P A P B . 2. Rule for Complements: The probability of the complement of an event A — P A (not ) —can be calculated as P A P A C ( ) 2 ( ) 1 or P A P A C ( ) 2 1 ( ) 3. The Multiplication Rule: The probability of an intersection of two events— P A B (both and ) —can be calculated as ∩ P A B P A P B A H20919 ( ) ( ) ( ) or P B P A B H20919 ( ) ( ) If A and B are independent events, then ( ) ( ) ( ) ∩ P A B P A P B . Need to Know... ? Copyright 2020 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. 4.4 Rules for Calculating Probabilities 155 The Basics Experiment I An experiment can result in one of five equally likely simple events, 1 E , 2 E ,. . . , 5 E . Events A , B , and C are defined as follows. Use these events to answer the questions in Exercises 1–6 . A E E P A ( ) 4 : , . 1 3 B E E E E P B ( ) 8 : , , , . 1 2 4 5 C E E P C ( ) 4 : , . 3 4 1. Find the probabilities associated with the following events by listing the simple events in each. a. A c b. ∩ A B c. ∩ B C d. ∪ A B e. ) BC f. ) A B g. ∪ ∪ A B C h. ( ) ∩ A B c 2. Use the definition of a complementary event to find these probabilities: a. ( ) P A c b. ( ) ( ) ∩ P A B c Do the results agree with those obtained in Exercise 1? 3. Use the definition of conditional probability to find these probabilities: a.
eBook - PDF- Prem S. Mann(Author)
- 2020(Publication Date)
- Wiley(Publisher)
Hence, the required probability is: P (in favor or neutral ) = 175 ____ 300 = .5833 EXAMPLE 4.29 Taking a Course in Ethics Calculating the Probability of the Union of Two Mutually Exclusive Events Consider the experiment of rolling a die once. What is the probability that a number less than 3 or a number greater than 4 is obtained? Solution Here, the event a number less than 3 is obtained happens if either 1 or 2 is rolled on the die, and the event a number greater than 4 is obtained happens if either 5 or 6 is rolled on the die. Thus, these two events are mutually exclusive as they do not have any common outcome and cannot happen together. Hence, their joint probability is zero. The marginal probabilities of these two events are: P (a number less than 3 is obtained) = 2 __ 6 P (a number greater than 4 is obtained) = 2 __ 6 The probability of the union of these two events is: P (a number less than 3 or a number greater than 4 is obtained) = 2 __ 6 + 2 __ 6 = .6667 EXAMPLE 4.30 Rolling a Die 4.5 Union of Events and the Addition Rule 177 Exercises Concepts and Procedures 4.66 Explain the meaning of the union of two events. Give one example. 4.67 How is the addition rule of probability for two mutually exclu- sive events different from the rule for two events that are not mutually exclusive? 4.68 Consider the following addition rule to find the probability of the union of two events A and B: P (A or B) = P (A) + P (B) − P (A and B) When and why is the term P (A and B) subtracted from the sum of P (A) and P (B)? Give one example where you might use this formula. 4.69 When is the following addition rule used to find the probability of the union of two events A and B? P ( A or B ) = P ( A) + P (B ) Give one example where you might use this formula. 4.70 Given that A and B are two mutually exclusive events, find P (A or B) for the following.
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