Probability Tree

3 Key excerpts on "Probability Tree"

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  eBook - PDF

  Reliability and Statistics in Geotechnical Engineering

  • Gregory B. Baecher, John T. Christian(Authors)
  • 2005(Publication Date)
  • Wiley

    (Publisher)

  As in any modeling activity, assumptions and simplifications are made at each step in conceptualizing the dam as a system and creating an event tree. Different analysts have different ways of defining events, different ways of linking events together, and different ways of estimating parameters and assigning probabilities to events. An event tree reflects a belief structure about a system, about the natural environment within which the system resides, and about the natural and human processes that affect performance. 20.1.1 Event trees An event tree is a graphical representation of the many chains of events that might result from some initiating event, a few of which, should they occur, would lead to system failure. As the number of events increases, the diagram fans out like the branches of a tree, suggesting the name (Figure 20.2). An event tree begins with some accident-initiating event (Figure 20.3). This might be a flood, an earthquake, human agency, an internal flaw, or something else. The analysis attempts to generate all possible subsequent events, and correspondingly, events that might follow the subsequent events, and so on. The event outcomes are represented as branches issuing from the chance node representing a particular event. This process follows until many chains of events are generated, some of which lead to failure, but most of which do not. A conditional probability is associated with each event, given all the events preceding in the tree. A joint probability of a chain of events is calculated by multiplying the conditional event probabilities along the chain. Summing the probabilities of all the chains of events that start from a single initiating event and lead to failure yields the total probability of failure due to that one initiating event. Summing the probabilities over all initiating events yields the total probability of system failure.

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  eBook - PDF

  Risk Analysis in Engineering and Economics

  • 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.

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  Understandable Statistics

  Concepts and Methods, Enhanced

  • Charles Henry Brase, Corrinne Pellillo Brase(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  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. 180 Chapter 4 ELEMENTARY PROBABILITY THEORY Tree diagrams help us display the outcomes of an experiment involving several stages. If we label each branch of the tree with an appropriate probability, we can use the tree diagram to help us compute the probability of an outcome displayed on the tree. One of the easiest ways to illustrate this feature of tree diagrams is to use an experiment of drawing balls out of an urn. We do this in the next example. T REE DIAGRAM AND PROBABILITY Suppose there are five balls in an urn. They are identical except for color. Three of the balls are red and two are blue. You are instructed to draw out one ball, note its color, and set it aside. Then you are to draw out another ball and note its color. What are the outcomes of the experiment? What is the probability of each outcome? SOLUTION: The tree diagram in Figure 4-10 will help us answer these questions. Notice that since you did not replace the first ball before drawing the second one, the two stages of the experiment are dependent. The probability associated with the color of the second ball depends on the color of the first ball. For instance, on the top branches, the color of the first ball drawn is red, so we compute the probabilities of the colors on the second ball accordingly. The tree diagram helps us organize the probabilities.

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