Mathematics
Disjoint and Overlapping Events
Disjoint events in mathematics refer to events that cannot occur simultaneously, while overlapping events can occur at the same time. In probability theory, disjoint events have no outcomes in common, whereas overlapping events share common outcomes. Understanding the distinction between disjoint and overlapping events is important for calculating probabilities and making predictions in various mathematical scenarios.
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9 Key excerpts on "Disjoint and Overlapping Events"
eBook - PDFPuzzles, Paradoxes, and Problem Solving
An Introduction to Mathematical Thinking
- Marilyn A. Reba, Douglas R. Shier(Authors)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
That is, the probability of exactly one head is obtained by adding the probabilities of two outcomes: a head occurring only on the first toss ( HT ), and a head occurring only on the second toss ( TH ). This simpli-fication is based on the Addition Rule of Chapter 12, which states that the probability of an event is obtained by adding together the probabilities of its constituent outcomes. What makes this Addition Rule valid is that the constituent outcomes are disjoint : they cannot happen together. More generally, if events E and F have no outcomes in common, then the probability Pr[ E or F ] can be easily calculated by summing their respective probabilities. Definition Disjoint Events Events E and F are disjoint if they both cannot happen simultaneously. In this case, the probability that either E or F occurs is given by Pr[ E or F ] = Pr[ E ] + Pr[ F ] . In terms of the decision tree, being disjoint means that the events E and F contain no com-mon outcomes. This difference between Disjoint and Overlapping Events is shown schemat-ically by the Venn diagrams in Figure 14.1. Events E and F are disjoint in (a) while they overlap in (b). E F E F (a) (b) FIGURE 14.1 : Disjoint and Overlapping Events The following examples illustrate the concepts of independence and disjointness. Example 14.1 Calculate the Probability of Rolling a Sum of 7 or 11 with a Pair of Dice A pair of (six-sided) dice is rolled. We would like to determine the probability that the sum of the numbers showing on the two dice equals 7 or 11. Such a calculation is important in certain casino games. (a) List the outcomes constituting the event E in which the face Independent Events and Disjoint Events 297 values sum to 7, and list the outcomes constituting the event F in which the face values sum to 11. (b) List the outcomes for the combined event G in which the face values sum to either 7 or 11. (c) Calculate the probability Pr[ G ] = Pr[ E or F ] and compare it with Pr[ E ] + Pr[ F ].
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 - PDFEntropy Demystified: The Second Law Reduced To Plain Common Sense
The Second Law Reduced to Plain Common Sense
- Arieh Ben-naim(Author)
- 2007(Publication Date)
- World Scientific(Publisher)
For random variables, “independent” and “uncorrelated” events are different con-cepts. For single events, the two concepts are identical. 40 Entropy Demystified We can calculate the following two conditional probabilities: Pr { of A / given B } = 1 / 3 > Pr { of A } = 1 / 6 Pr { of A / given C } = 0 < Pr { of A } = 1 / 6 In the first example, the knowledge that B has occurred increases the probability of the occurrence of A . Without that knowledge, the probability of A is 1 / 6 (one out of six possibili-ties). Given the occurrence of B , the probability of A becomes larger , 1 / 3 (one out of three possibilities). But given that C has occurred, the probability of A becomes zero, i.e., smaller than the probability of A without that knowledge. It is important to distinguish between disjoint (i.e., mutu-ally exclusive events) and independent events. Disjoint events are events that are mutually exclusive; the occurrence of one excludes the occurrence of the second. Being disjoint is a prop-erty of the events themselves (i.e., the two events have no com-mon elementary event). Independence between events is not defined in terms of the elementary events comprising the two events, but in terms of their probabilities. If the two events are disjoint, then they are strongly dependent . The following example illustrates the relationship between dependence and the extent of overlapping. Let us consider the following case. In a roulette, there are altogether 12 numbers { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 } Each of us chooses a sequence of six consecutive numbers, say, I choose the sequence: A = { 1, 2, 3, 4, 5, 6 } and you choose the sequence: B = { 7, 8, 9, 10, 11, 12 } The ball is rolled around the ring. We assume that the roulette is “fair,” i.e., each outcome has the same probability Introduction to Probability Theory, Information Theory, and all the Rest 41 Fig.
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- Dawn A. Willoughby(Author)
- 2016(Publication Date)
- Wiley(Publisher)
The sample space of the experiment is represented by a rectangle and each event is shown as a circle or an ellipse within the rectangle. 286 A N E S S E N T I A L G U I D E T O B U S I N E S S S T A T I S T I C S In its simplest form, a Venn diagram can be drawn to display a single event. In the diagram below, the event A is represented by the shaded area inside the ellipse. S A Venn diagrams are particularly useful for showing the relationships between two or more events and we will use them in the remainder of this chapter as new concepts are introduced. Mutually Exclusive Events Events that cannot occur at the same time are known as mutually exclusive events; they have no common elements. Each time an experiment is repeated, no more than one of a collection of mutually exclusive events will occur. The Venn diagram below provides a visual representation of two mutually exclusive events, A and B. S A B Suppose we conducted an experiment that involved rolling a die and recording the number facing upwards. Defining two events, A: ‘the number facing upwards is odd’, and B: ‘the number facing upwards is even’, we can see that the events A and B are mutually exclusive because they cannot occur at the same time: the number facing upwards on a die cannot be odd and even. There are no shared elements in events A and B: A f1; 3; 5g, whereas B f2; 4; 6g. Independent Events Two events are called independent events if the occurrence of one event in an experiment does not have any effect on the occurrence of the other event. Suppose we conducted an experiment that involved rolling a die, recording the number facing upwards and flipping a coin, recording the side facing upwards. Defining two events, A: ‘the number facing upwards is odd’, and B: ‘the side facing upwards is a head’, we can see that the events A and B are independent because the probability of getting an odd number facing upwards on the die does not influence the probability of the coin side being a ‘head’.
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.
- Andrei D. Polyanin, Alexander V. Manzhirov(Authors)
- 2006(Publication Date)
- Chapman and Hall/CRC(Publisher)
The elementary outcomes of A are the elementary outcomes that do not belong to the event A . An event is said to be sure if it necessarily occurs for each realization of the condition set Σ . Obviously, the sure event is equivalent to the space of elementary events, and hence the sure event should be denoted by the symbol Ω . An event is said to be impossible if it cannot occur for any realization of the condition set Σ . Obviously, the impossible event does not contain any elementary outcome and hence should be denoted by the symbol ∅ . Two events A and A are said to be opposite if they simultaneously satisfy the following two conditions: A ∪ A = Ω , A ∩ A = ∅ . Events A and B are said to be incompatible , or mutually exclusive , if their simultaneous realization is impossible, i.e., if A ∩ B = ∅ . Events H 1 , . . . , H n are said to form a complete group of events , or to be collectively exhaustive , if at least one of them necessarily occurs for each realization of the condition set Σ , i.e., if H 1 ∪ · · · ∪ H n = Ω . 1031 1032 P ROBABILITY T HEORY Events H 1 , . . . , H n form a complete group of pairwise incompatible events (are hy-potheses ) if exactly one of the events necessarily occurs for each realization of the condition set Σ , i.e., if H 1 ∪ · · · ∪ H n = Ω and H i ∩ H j = ∅ ( i ≠ j ). Main properties of random events: 1. A ∪ B = B ∪ A and A ∩ B = B ∩ A (commutativity). 2. ( A ∪ B ) ∩ C = ( A ∩ C ) ∪ ( B ∩ C ) and ( A ∩ B ) ∪ C = ( A ∪ C ) ∩ ( B ∪ C ) (distributivity). 3. ( A ∪ B ) ∪ C = A ∪ ( B ∪ C ) and ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (associativity). 4. A ∪ A = A and A ∩ A = A . 5. A ∪ Ω = Ω and A ∩ Ω = A . 6. A ∪ A = Ω and A ∩ A = ∅ . 7. ∅ = Ω , Ω = ∅ , and A = A ; 8. A B = A ∩ B . 9. A ∪ B = A ∩ B and A ∩ B = A ∪ B ( de Morgan’s laws ).
eBook - PDFFarewell To Entropy, A: Statistical Thermodynamics Based On Information
Statistical Thermodynamics Based on Information
- Arieh Ben-naim(Author)
- 2008(Publication Date)
- World Scientific(Publisher)
In mathematical terms, we assume (1) A i · A j = φ for each pair of events i, j i = j = 1 , 2 , . . . , n , (2.6.1) (2) Ω = ∑ n i =1 A i (or n i =1 A i ) (2.6.2) 66 Statistical Thermodynamics Based on Information Example: Consider the outcomes of throwing a die. Define the events A 1 = { even outcome } = { 2 , 4 , 6 } , A 2 = { odd and larger than one } = { 3 , 5 } , A 3 = { 1 } . (2.6.3) Clearly, these three events are disjoint and their union covers the entire range of possible outcomes. Exercise : Show that for any n events A 1 , A 2 , . . . A n ; if they are pairwise mutually exclusive (i.e., A i · A j = φ for each pair of i = j ), then it follows that each group of events are mutually exclusive. The inverse of this statement is not true. For instance, events A, B and C can be mutually exclusive (i.e., A · B · C = φ ) but not pairwise mutually exclusive. For the second example, we use again the squared board of unit area and any division of the area into several mutually exclusive areas, as in Fig. 2.9a. Clearly, by construction, each pair of events consists of mutually exclusive events, and the sum of the events is the certain event, i.e., hitting any place on the board (neglecting the borderlines) is unity. Let B be any event. We can always write the following equalities: B = B · Ω = B · n i =1 A i = n i =1 B · A i . (2.6.4) The first equality is evident (for any event B , its intersection with the total space of events Ω, gives B ). The second equality follows from the assumption (2.6.2). The third equality follows from the distributive law of the product of events. 13 From the assumption that all A i are mutually exclusive, it also follows that all B · A i are mutually exclusive. 14 Therefore, it follows that the probability of B is the sum of the probabilities of all the 13 Show that A · ( B + C ) = A · B + A · C for any three events A, B, C and by generalization B · P A i = P B · A i .
eBook - PDF- Kushwaha, K.S.(Authors)
- 2021(Publication Date)
- NEW INDIA PUBLISHING AGENCY (NIPA)(Publisher)
if A B and B C Q.110 A and B are disjoint (mutually exclusive) if A B = ϕ (null set). Q.111 We have A B = A + B if A and B are disjoint. Q.112 Since all the events are subsets of S, all the laws of set theory hold for algebra of events. Q.113 Atleast one of the events A or B occurs → w є A B Q.114 Neither A nor B occurs w є A B Q.115 Event A occurs and B does not occur w є A B Q.116 Probability of complementary event A of A is gfiven by P ( A ) = 1 - P (A) Q.117 P ( A B ) = P ( B ) - P ( A B ) Q.118 If B A then P ( A B ) = P(A) - P(B) Q.119 If A and B are two events and are not disjoints then P(A B) = P(A) + P(B) - P(A B) Q.120 For three non mutually exclusive events A, B & C P(A B C) = P(A) + P(B) + P(C)-P(A B)-P(B C) -P (C A) +P(A B C) Q.121 The probabilities P (A l ), P (A 2 ), ...... , P (A n ) of mutually exclusive forms of A, are known as “partial probabilities” U > U > U > 87 Q.122 According to Boole’s inequality for n events A1, A2, ...... , A n , we have Q.123 Q.124 For n events A1, A2, ...... , A n Q.125 For two events A and B, we have P(A B)= P(A).P(B / A),P(A) >0 or = P(B).P(A / B),P(B) > 0 where P (B/ A) represents conditional probability of occurence of B when the event A has already happend. Q.126 An event A is said to be independent of another event B if P (A/B) = P (A) if P (B) ≠ 0. Q.127 If the events A and B are such that P (A) ≠ 0, P (B) ≠ 0 and A is independent of B then B is independent of A if P(A B) = P(A)p(B) Q.128 For a fixed event B, with P (A) ≠ 0 we have Q.129 P(A B / C) = P(A / C)+ P (B / C)-P(A B / C) Q.130 If A and B are independent events then A and B , A and B, and A & B are also independent. Q.131 Pair wise Independent Events : The events A1, A2, ......... , A n are said to be pair wise independent if P ( A i A j ) = P(Ai)P(Aj), i ≠ j = 1,2, ....... , n 88 Q.132 Mutually Independent Events : The events A1, A2, .......... , A n are said to be mutually independent if P ( A i1 A j2 A i3 ........ A ik ) = P ( A i1 ) P ( A j2 ) ......
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