Mathematics
Mutually Exclusive Probabilities
Mutually exclusive probabilities refer to events that cannot occur simultaneously. In other words, if one event happens, the other cannot. When calculating the probability of mutually exclusive events, the probabilities of each event are added together. For example, when rolling a standard six-sided die, the probabilities of rolling a 2 or a 4 are mutually exclusive.
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eBook - PDFMathematics NQF4 SB
TVET FIRST
- M Van Rensburg, I Mapaling M Trollope(Authors)
- 2017(Publication Date)
- Macmillan(Publisher)
Dependent events When the occurrence of one event affects the probability of the other event occurring The probability of drawing an ace from a deck of cards is 4 __ 54 . If you drew an ace and did not put it back in the deck, the next time you draw a card, the probability of drawing an ace will be 3 __ 53 . So, the probability of drawing an ace a second time is depen-dent on the previous event. Mutually exclusive events When the occurrence of one event makes it impossible for the other event to occur or have a successful outcome When you are 21 years old (A), you cannot be 25 (B) at the same time. There is no intersection between events A and B: S A B A ∩ B = 0 Remember: There are four suits in a deck of cards: spades, diamonds, clubs and hearts. Each of the four suits contains the following 13 cards: In addition to the above, a deck also includes two Joker cards. 356 Module 13 Term Description Example Mutually inclusive events When both events can occur or have success-ful outcomes at the same time When a trial tests the proba-bility of drawing an ace (A) or a red card (B) from a deck of cards, events A and B are mutually inclusive, because you can draw a card that is both an ace and red, which would occur in the intersec-tion of events A and B. S A B Comple-mentary events All the events in the sample space, except the given event; in other words, all the events that will NOT include the given event When you toss a coin, the only possible outcomes are ‘heads’ (H) or ‘tails’ (T). Because there are only two possible outcomes, the proba-bility of NOT getting ‘heads’ ( H ̅ ), is the same as the proba-bility of getting ‘heads’ (H): S H H The complement of H is H ̅ . Unit 13.2: Predictions based on validated experimental probabilities Before we make predictions based on experimental probabilities, let us look at the properties of probability. Note: P(not H) = p( H ̅ ) Note: We can express probability as a fraction, a decimal or a percentage.
eBook - PDF- L. Z. Rumshiskii(Author)
- 2016(Publication Date)
- Pergamon(Publisher)
§ 2 . THE CLASSICAL D E F I N I T I O N OF P R O B A B I L I T Y Let us first of all agree on some notation. Events are called mutually exclusive if they cannot occur simultaneously. A collec-tion of events form a partition if at each trial one and only one of the events must occur; i.e. if the events are pair-wise mutually exclusive and if only one of them occurs. In this section we restrict our attention to trials with equally likely outcomes', for example we shall consider the throwing of an unbiased die with possible outcomes 1, 2, 3, 4, 5 and 6.f In other words we shall study trials whose possible outcomes can be re-presented by a partition of equally likely events ; in these circum-stances the events will be known as cases. If the partition consists of Ν equally likely cases, then each case will have probability equal to l/N. This accords with the fact that in a large number of trials equally likely cases occur approximately t We shall not try to define the idea of equally likely outcomes in terms of any simpler concepts. It is usually based on some consideration of symmetry, as in the example of the die, and connected in practice with the approximate equality of the relative frequencies of all the outcomes in a large number of trials. We remark that everywhere in this section we shall assume that we are dealing with a finite number of cases. E V E N T S A N D P R O B A B I L I T I E S 3 the same number of times, i.e. they have relative frequencies near to l/N. For example, in throwing an unbiased die the cases are the appearance of 1, 2, 3, 4, 5, and 6 points, and these form a partition; each case will have probability equal to Let us consider now a compound event A comprising M cases. The probability of the event A is defined to be MjN.
No longer available |Learn more- Anthony Hayter(Author)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
FIGURE 1.52 Three mutually exclusive events A B C S Copyright 2011 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. 30 CHAPTER 1 PROBABILITY THEORY Union of Mutually Exclusive Events For a sequence A 1 , A 2 ,..., A n of mutually exclusive events , the probability of the union of the events is given by P ( A 1 ∪ ··· ∪ A n ) = P ( A 1 ) + ··· + P ( A n ) If a sequence A 1 , A 2 ,..., A n of mutually exclusive events has the additional property that their union consists of the whole sample space S , then they are said to be an exhaustive sequence. They are also said to provide a partition of the sample space. Sample Space Partitions A partition of a sample space is a sequence A 1 , A 2 ,..., A n of mutually exclusive events for which A 1 ∪ ··· ∪ A n = S Each outcome in the sample space is then contained within one and only one of the events A i . Figure 1.53 illustrates a partition of a sample space S into eight mutually exclusive events. Example 5 Television Set Quality In addition to the events A and B discussed before, consider also the event C that an appliance is of “mediocre quality.” The event is defined to be appliances that score either Satisfactory or Good on each of the two evaluations, so that C = { ( S , S ), ( S , G ), ( G , S ), ( G , G ) } The three events A , B , and C are illustrated in Figure 1.54.
eBook - PDF- Barbara Illowsky, Susan Dean(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
In sampling without replacement, each member of a population may be chosen only once, and the events are considered not to be independent. When events do not share outcomes, they are mutually exclusive of each other. 3.3 Two Basic Rules of Probability The multiplication rule and the addition rule are used for computing the probability of A and B, as well as the probability of A or B for two given events A, B defined on the sample space. In sampling with replacement each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered to be not independent. The events A and B are mutually exclusive events when they do not have any outcomes in common. 3.4 Contingency Tables There are several tools you can use to help organize and sort data when calculating probabilities. Contingency tables help display data and are particularly useful when calculating probabilites that have multiple dependent variables. 3.5 Tree and Venn Diagrams A tree diagram use branches to show the different outcomes of experiments and makes complex probability questions easy to visualize. A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events. A Venn diagram is especially helpful for visualizing the OR event, the AND event, and the complement of an event and for understanding conditional probabilities. FORMULA REVIEW 3.1 Terminology A and B are events P(S) = 1 where S is the sample space 0 ≤ P(A) ≤ 1 P(A|B) = P( AANDB) P(B) 3.2 Independent and Mutually Exclusive Events If A and B are independent, P(A AND B) = P(A)P(B), P(A|B) = P(A) and P(B|A) = P(B).
eBook - PDF- Kathleen Subrahmaniam(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
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 }. Basic Concepts of Probability 13 Two events which cannot occur at the same time are said to be mutually exclusive (m. e .) . We would speak of the sets corresponding to these events as being disjoint since they have no points in common. We use the symbol 0 to designate the null set. If the events A^, A2, . . . , A^ are mutually exclusive ( i .e ., the inter-section of any pair is 0) and exhaustive ( i .e ., their union is S), then we say that these k events form a partition of the sample space. Note that the simple events which describe the sample space always form a partition of S. Often we may be interested in the fact that a particular event has not occurred. In passing we note that A and A are m .e» and form a partition of S since A U A = S. In discussing events it will often be helpful to decompose an event into the union of m .e . components. Suppose the events A^, A2, . ••, A^ form a partition of S. Then any event F in S can be written as f = (Ax n F) u (A 2 n f> u ••• u (AR n f> . To illustrate this concept, consider the following Venn diagram in which Blt B2, B 3 form a partition of S. 14 Chapter 2 F = (Bi n F) u (b 2 n f ) u (b 3 n f> . Note that in this case the events B 3 and F have no points in common; thus B 3 fl F = 0. Now consider the special case of the event Bx and its com ple-ment Br Since B1 and Bj form a partition of S, we can write F = (B jflF ) U (Bi H F ). In the above drawing, 1^ = 6 2 U B 3. To illustrate this result, consider a bolt-manufacturing plant. The bolts are produced in three shifts: 8 to 4, 4 to 12, and 12 to 8 .
eBook - PDF- D.G. Rees(Author)
- 2018(Publication Date)
- Chapman and Hall/CRC(Publisher)
Applying the special case of the law of addition, P iE , or E or...or En) = P(iq) + P(P2) + ... + P(En) But since the events are exhaustive, one of them must occur, and so the left-hand side of the equation is 1. In other words, then, The sum o f the probabilities of a set of m utually exclusive and exhaustive events is 1. Probability ■ 57 This result is useful in checking whether we have correctly calculated the separate probabilities of the various mutually exclusive events of an experiment. Probability Example 5.8 For families with two children, what are the probabilities of the various possibilities, assuming that boys and girls are equally likely at each birth? Four mutually exclusive and exhaustive events are BB, BG, GB, and GG, where BG means a boy followed by a girl. Therefore, using the special law of multiplication. Similarly, P(BB) = P(GB) = P(GG) l = 4 , must occur, and the total probability is 1 . 5.12 Complementary Events and the Calculation of P (at Least 1 ...) For any event E, there is a com plem entary event E r which we call not E. Since either E or E' must occur, and they cannot both occur, P (E )+ P (E r) = 1, which is a special case of the result of the previous section. It follows that: P(E) = 1 -P(E'). This result is useful in some cases, where it is easier to calculate the probability of the com plem ent of some event and subtract the answer from 1 than it is to calculate the probability of the event directly. This is especially true when we wish to calculate the probability that ‘at least 1 of something will occur in a number of trials’, since: P (at least 1...) = 1 —P(none...) (5.7) Probability Example 5.9 For families with four children, what is the probability that there will be at least one boy, assuming boys and girls are equally likely? Instead of
eBook - PDF- Prem S. Mann(Author)
- 2016(Publication Date)
- Wiley(Publisher)
Thus, these two events are mutually exclusive as they do not have any common outcome and Calculating the probability of the union of two mutually exclusive events. Addition Rule to Find the Probability of the Union of Mutually Exclusive Events The probability of the union of two mutually exclusive events A and B is P( A or B) = P( A) + P( B) 4.5 Union of Events and the Addition Rule 161 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 is obtained or a number greater than 4 is obtained) = 2 6 + 2 6 = .6667 EXERCISES 4.73 Two thousand randomly selected adults were asked whether or not they have ever shopped on the Internet. The following table gives a two-way classification of the responses. Have Shopped Have Never Shopped Male 500 700 Female 300 500 Suppose one adult is selected at random from these 2000 adults. Find the following probabilities. a. P(has never shopped on the Internet or is a female) b. P(is a male or has shopped on the Internet) c. P(has shopped on the Internet or has never shopped on the Internet) 4.74 Two thousand randomly selected adults were asked if they think they are financially better off than their parents. The following table gives the two-way classification of the responses based on the educa- tion levels of the persons included in the survey and whether they are financially better off, the same as, or worse off than their parents. Less Than High School High School More Than High School Better off 140 450 420 Same 60 250 110 Worse off 200 300 70 Suppose one adult is selected at random from these 2000 adults. Find the following probabilities. a. P(better off or high school) b.
- Lawrence Kupper, Brian Neelon, Sean M. O'Brien(Authors)
- 2010(Publication Date)
- Chapman and Hall/CRC(Publisher)
4 Basic Probability Theory (ii) The probability of the union of p events is given by: pr ∪ p i = 1 A i = p i = 1 pr ( A i ) − p − 1 i = 1 p j = i + 1 pr ( A i ∩ A j ) + p − 2 i = 1 p − 1 j = i + 1 p k = j + 1 pr ( A i ∩ A j ∩ A k ) − · · · + ( − 1 ) p − 1 pr ∩ p i = 1 A i . As important special cases, we have, for p = 2, pr ( A 1 ∪ A 2 ) = pr ( A 1 ) + pr ( A 2 ) − pr ( A 1 ∩ A 2 ) and, for p = 3, pr ( A 1 ∪ A 2 ∪ A 3 ) = pr ( A 1 ) + pr ( A 2 ) + pr ( A 3 ) − pr ( A 1 ∩ A 2 ) − pr ( A 1 ∩ A 3 ) − pr ( A 2 ∩ A 3 ) + pr ( A 1 ∩ A 2 ∩ A 3 ) . 1.1.2.2 Mutually Exclusive Events For i = j , two events A i and A j are said to be mutually exclusive if these two events cannot both occur (i.e., cannot occur together) when the experiment is conducted; equivalently, the events A i and A j are mutually exclusive when pr ( A i ∩ A j ) = 0. If the p events A 1 ,A 2 , . . . ,A p are pairwise mutually exclusive, that is, if pr ( A i ∩ A j ) = 0 for every i = j , then pr ∪ p i = 1 A i = p i = 1 pr ( A i ) , since pairwise mutual exclusivity implies that any intersection involving more than two events must necessarily have probability zero of occurring. 1.1.2.3 Conditional Probability For i = j , the conditional probability that event A i occurs given that (or con-ditional on the fact that) event A j occurs when the experiment is conducted, denoted pr ( A i | A j ) , is given by the expression pr ( A i | A j ) = pr ( A i ∩ A j ) pr ( A j ) , pr ( A j ) > 0. Concepts and Notation 5 Using the above definition, we then have: pr ∩ p i = 1 A i = pr A p | ∩ p − 1 i = 1 A i pr ∩ p − 1 i = 1 A i = pr A p | ∩ p − 1 i = 1 A i pr A p − 1 | ∩ p − 2 i = 1 A i pr ∩ p − 2 i = 1 A i . . . = pr A p | ∩ p − 1 i = 1 A i pr A p − 1 | ∩ p − 2 i = 1 A i · · · pr ( A 2 | A 1 ) pr ( A 1 ) . Note that there would be p ! ways of writing the above product of p probabilities.
eBook - PDF- Prem S. Mann(Author)
- 2017(Publication Date)
- Wiley(Publisher)
Either the loan application will be approved or it will be rejected. Hence, P( A and R) = 0 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. Joint Probability of Mutually Exclusive Events The joint probability of two mutually exclusive events is always zero. If A and B are two mutually exclusive events, then, P( A and B) = 0 EXERCISES 4.51 Given that P ( B ) = .65 and P ( A and B ) = .45, find P ( A ∣ B ). 4.52 Given that P ( A ∣ B ) = .40 and P ( A and B ) = .36, find P ( B ). 4.53 Given that P ( B ∣ A ) = .80 and P ( A and B ) = .58, find P ( A ). APPLICATIONS 4.54 Five hundred employees were selected from a city’s large pri- vate companies and asked whether or not they have any retirement benefits provided by their companies. Based on this information, the following two-way classification table was prepared. Have Retirement Benefits Yes No Men 240 60 Women 135 65 a. Suppose one employee is selected at random from these 500 employees. Find the following probabilities. i. Probability of the intersection of events “woman” and “yes” ii. Probability of the intersection of events “no” and “man” CONCEPTS AND PROCEDURES 4.44 What is meant by the intersection of two events? Give one example. 4.45 Briefly explain the meaning of the joint probability of two or more events. Give one example. 4.46 How is the multiplication rule of probability for two dependent events different from the rule for two independent events? 4.47 Briefly explain the meaning of the joint probability of two mutually exclusive events. 4.48 Find the joint probability of A and B for the following. a. P ( A ) = .40 and P ( B ∣ A ) = .25 b. P ( B ) = .65 and P ( A ∣ B ) = .36 4.49 Given that A and B are two independent events, find their joint probability for the following.
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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