Mathematics
Tree Diagram
A tree diagram is a visual tool used to represent possible outcomes and decision paths in a systematic way. It consists of branches that extend from a central node, with each branch representing a different option or outcome. Tree diagrams are commonly used in probability and decision analysis to illustrate and calculate the likelihood of various events occurring.
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5 Key excerpts on "Tree Diagram"
eBook - PDFFinite Mathematics
Models and Applications
- Carla C. Morris, Robert M. Stark(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
⧫ While Tree Diagrams can be popular, visual representations do not rely solely on them. For larger numbers in practical problems, Tree Diagrams can become impractical. Still, they have a role (sometimes, wall sized) in planning large construction projects. In the space program and other large programs, they are adapted to coordinate a multi- tude of activities. EXERCISES 7.3 1. Find all subsets of A = {X, Y, Z}. 2. Find all subsets of A = {1, 2, 3, 4}. 220 SET AND PROBABILITY RELATIONSHIPS 3. An experiment consists of drawing a ball from an urn that contains three yellow, two white, one red, and four green balls. After the color is noted, a second ball is drawn (without replacement) and the color is again noted. Use a Tree Diagram to determine the number of outcomes. 4. An inspector rates four items as “excellent,” as “satisfactory,” or as “unacceptable.” How many possible reports can the inspector make? 5. In a psychology experiment, five rats are placed in a cage. Two rats are trained and three untrained. A rat is selected and labeled T if it is trained and U if not. The rat is placed in another cage and cannot be selected again. Use a Tree Diagram to describe the experiment when three rats are selected. 6. How many different four digit numbers less than 6500 can be formed from the digits {3, 6, 8, 9} a) if digits can be used repeatedly? b) if digits cannot repeat? Answer Exercises 7–11 using Tree Diagrams. 7. Exercise 7.2.8. 8. Exercise 7.2.9. 9. Exercise 7.2.10. 10. Exercise 7.2.11. 11. Exercise 7.2.12. 12. A coin is flipped until either of two heads appears or four flips have occurred. Use a Tree Diagram to list the possible outcomes. 13. A die is tossed and its face is noted. If it is an odd number, a coin is flipped. If it is an even number, a letter (a or b or c) is randomly selected. Use a Tree Diagram to depict possible outcomes. 14. A quality control inspector tests the items produced on an assembly line and rates them as G (good) or D (defective).
eBook - PDF- Bilal M. Ayyub(Author)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
3.3.5.5 Tree Construction Decision trees are commonly used to structure and analytically examine the available information on a decision situation. Risk profiles could provide some of the necessary information. A decision tree includes the decision and chance nodes. The decision nodes, which are represented by squares in a decision tree, are followed by possible actions (or alternatives, A i ) that can be selected by a decision maker. The chance nodes, which are represented by circles in a decision tree, are followed by outcomes that can occur without the complete control of the decision maker. The outcomes have both probabilities ( P ) and consequences ( C ). Here, the consequence can be a cost. Each tree segment followed from the beginning (left end) of the tree to the end (right end) of the tree is called a branch . Each branch represents a possible scenario of decisions and possible outcomes, and the total expected consequence (cost) for each branch can be computed. Then, the most suitable decisions can be selected to obtain the minimum cost. In general, utility values can be used and maximized instead of cost values. Also, decisions can be based on risk profiles by considering both the total expected utility value and the standard deviation of the util-ity value for each alternative. The standard deviation can be critical for decision making as it provides a measure of uncertainty for the utility values of the alternatives as discussed in Chapter 7. Influence diagrams can be constructed to model dependencies among deci-sion variables, outcomes, and system states using the same symbols of Figure 3.8.
- Susanna Epp(Author)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
677 CHAPTER 10 Throughout the book graphs and trees have been used as convenient visualizations. For in-stance, a graph can model a wide variety of structures such as the arrangement of electric power lines or fiber optic cables, a transportation system, a knowledge base, or a collection of computers ranging from a small local area network to the entire world wide web. A graph model can be used to solve a logical problem, color a map, or schedule meetings. A possibil-ity tree shows all potential results of a multistep operation with a finite number of outcomes for each step, the directed graph of a relation on a set shows which element of the set are related to which, a Hasse diagram illustrates the relations among elements in a partially or-dered set, and a PERT diagram shows which tasks must precede which in executing a project. In Chapter 1 we introduced the basic terminology of graphs, and in Section 4.9 we used properties of even and odd integers and direct and indirect proof to prove the handshake theorem and derive some of its consequences. We first proved the formula for the number of edges in a complete graph on n vertices using the handshake theorem, and then reproved it using mathematical induction in Section 5.3, recursion in Section 5.6, and combinatorial reasoning in Section 9.5. In this chapter we go more deeply into the mathematics of graphs and trees by exploring the concepts of connectedness, Euler and Hamiltonian circuits, representation of graphs by matrices, isomorphisms of graphs, the relations between the number of vertices and the number of edges in a tree, properties of rooted trees, spanning trees, and finding short-est paths in graphs. Applications include uses of graphs and trees in the study of decision problems, chemistry, data storage, computer language syntax, and transportation networks. Trails, Paths, and Circuits One can begin to reason only when a clear picture has been formed in the imagination.
eBook - PDFDiscrete Mathematics
Mathematical Reasoning and Proof with Puzzles, Patterns, and Games
- Douglas E. Ensley, J. Winston Crawley(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
7 Graphs and Trees The story of graph theory is a fascinating testimony to the study of recreational mathematics. Many early results in the area apply to nothing more than puz- zles on maps or chessboards, and yet today graph theory comprises some of the fastest growing branches of applied mathematics. Graph theoretical concepts are used to design computer circuits, create production schedules, optimize communication networks, and countless other modern tasks. In the realm of abstract mathematics, graphs are often used to add visualization or simplification to difficult concepts. In fact, we have already seen examples of graphs and trees in earlier sections of this book: • In determining all possible outcomes of a “best-of-three” match between Players A and B, we visualized all possible matches using a “game tree” as in Figure 7-1. • We have also used “arrow diagrams” to help us visualize properties of func- tions and relations as in Figure 7-2. All these diagrams are examples of graphs. The first is a special type of graph called a tree. The second and third are called directed graphs to emphasize the role of the arrows in the diagrams. In this chapter, not only will we see what these various diagrams have in common, but we will also study a number of additional applications of graphs and trees. A B A B A B A B A B AA ABA ABB BAA BAB BB • • • • • • • • • • • Figure 7-1 Game tree for best-of-three series. 4 3 2 1 4 3 2 1 3 4 1 2 • • • • • • • • • • • • Figure 7-2 Arrow diagrams for functions and relations. 505 506 Chapter 7 / Graphs and Trees 7.1 Graph Theory Origins and Euler The origin of graph theory is usually traced to a paper written by the great Swiss mathematician Leonhard Euler (1707–1783).
eBook - PDF- Paul Goodwin, George Wright(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
The implications of this result are far reaching. Decision trees constructed early in the analyst/decision-maker interaction may be incomplete repre- sentations of the decision problem facing the decision maker. Eliciting decision-tree representations What methods have been developed to help elicit decision-tree representations from decision makers? One major method, much favored by some decision analysts, is that of influence diagrams 13 which are designed to summarize the dependencies that are seen to exist among events and acts within a decision. Such dependencies may be me- diated by the flow of time, as we saw in our examples of decision trees. As we shall see, a close relationship exists between influence diagrams and the more familiar de- cision trees. Indeed, given certain conditions, influence diagrams can be converted to trees. The advantage of starting with influence diagrams is that their graphic represen- tation is more appealing to the intuition of decision makers who may be unfamiliar with decision technologies. In addition, influence diagrams are more easily revised and altered as the decision maker iterates with the decision analyst. Decision trees, because of their strict temporal ordering of acts and events, need respecifying com- pletely when additional acts and events are inserted into preliminary representations. We shall illustrate the applicability of influence diagrams through a worked example. First, however, we will present the basic concepts and representations underlying the approach. Figure 7.11 presents the key concepts. As with the decision tree, event nodes are represented by circles and decision nodes by squares. Arrowed lines between nodes indicate the influence of one node on another. For example, an arrow pointing to an event node indicates that the likelihood of events (contained in the node) is influenced by either a prior decision or the occurrence (or not) of prior events.
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