The purpose of this thesis was to thoroughly develop probability objectives to be used by a fifth grade teacher. These probability objectives were developed in four main units. The focus of the first unit was probability vocabulary. The second unit explored the concept of fairness as determined by the probability of winning a game. The third unit's purpose was to determine sample space using tree diagrams, lists, and the Counting Principle. Unit four was designed to help the student write theoretical and experimental probability as fractions, decimals, and percentages. Each unit was written to include detailed descriptions, definitions, and probability activities that can be used in a fifth grade classroom.

In this work we present a novel computational framework for analyzing evolution of uncertainty in state trajectories of a hypersonic air vehicle due to uncertainty in initial conditions and other system parameters. The framework is built on the so called generalized Polynomial Chaos expansions. In this framework, stochastic dynamical systems are transformed into equivalent deterministic dynamical systems in higher dimensional space. In the research presented here we study evolution of uncertainty due to initial condition, ballistic coefficient, lift over drag ratio and atmospheric density. We compute the statistics using the continuous linearization (CL) approach. This approach computes the jacobian of the perturbational variables about the nominal trajectory. The covariance is then propagated using the riccati equation and the statistics is compared with the Polynomial Chaos method. The latter gives better accuracy as compared to the CL method. The simulation is carried out assuming uniform distribution on the parameters (initial condition, density, ballistic coefficient and lift over drag ratio). The method is then extended for Gaussian distribution on the parameters and the statistics, mean and variance of the states are matched with the standard Monte Carlo methods. The problem studied here is related to the Mars entry descent landing problem.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / O presente trabalho tem por finalidades: demonstrar o problema da agulha de Buffon, fazer uma pequena generalizaÃÃo do resultado obtido e apresentar aplicaÃÃes baseadas nos fundamentos do referido problema. O problema da agulha de Buffon està inserido no estudo da Teoria das Probabilidades, particularmente na subÃrea de probabilidade geomÃtrica. Para chegarmos à soluÃÃo desta questÃo, alÃm dos conceitos e propriedades atinentes à Teoria das probabilidades à necessÃrio o conhecimento de noÃÃes bÃsicas do cÃlculo integral. Nos capÃtulos 2, 3 e 4 à apresentado um estudo preliminar sobre probabilidade, com os conceitos bÃsicos, propriedades e a formulaÃÃo de alguns modelos probabilÃsticos. Durante o desenvolvimento do trabalho, sempre que possÃvel, os conceitos e definiÃÃes sÃo inseridos com o auxÃlio de um problema motivador e para fixaÃÃo dos mesmos sÃo mostrados exemplos resolvidos. O Ãltimo capÃtulo evidencia a importÃncia do problema de Buffon como mÃtodo para realizar estimativas e como fundamento para o processo de captaÃÃo de imagens pelos aparelhos de tomografia computadorizada, um grande avanÃo para a Medicina no que diz respeito ao diagnÃstico por imagens. / This paper has the objective of showing Buffon's needle problem, doing a minor generalization of the results obtained hereby, and also presenting some applications based upon the fundamentals of such problem. Buffon's needle problem has been inserted into the study of Theory of Probability, particularly in its sub-area of geometrical probability. In order to attain the solution to this question, in addition to the concepts and the properties concerning the theory of probabilities, it is necessary that one should have some basic knowledge about integral calculus. In chapters 2, 3, and 4 there is a preliminary study of probability, with the basic concepts, properties and formulation of some probabilistic models being presented. During the development of this paper, whenever it was possible, the concepts and definitions were inserted with the aid of a motivational problem and they were solved by means of fixing the same examples as shown. The final chapter presents the importance of Buffon's needle problem as a method of making estimates and as a foundation for the process of capturing images in CT (computerized tomography) scanning machines, such a great breakthrough in what concerns the diagnosis by means of imaging.

This paper presents an innovat ive cons t ruct ion of a probabilistic model for predicting chance situations. It describes the construction of a four phase model, derived from an intense qualitative analysis of the written responses of 94 mathematically talented middle school students to the probabilistic compound event problem: “How many doubles are expected when rolling two dice fifty times?” We found that the students’ comprehension process of compound event situations can be broken down into a four phase model: beliefs, subjective estimations, chance estimations and probabilistic calculations. The paper focuses on the development of the model over the course of the experiment, identifying the process the students underwent as they attempted to answer the question. We explain each phase as it was reflected in the students\' rationalizations. All phases, including their definitions and students’ citations, will be presented in the paper. While not every student necessarily goes through all four phases, an awareness and understanding of them all allows for efficient, effective intervention during the learning process. We found that guidance and learning intervention helped shorten the preliminary phases, leading to more relative time spent on probabilistic calculations.

This paper presents an innovat ive cons t ruct ion of a probabilistic model for predicting chance situations. It describes the construction of a four phase model, derived from an intense qualitative analysis of the written responses of 94 mathematically talented middle school students to the probabilistic compound event problem: “How many doubles are expected when rolling two dice fifty times?” We found that the students’ comprehension process of compound event situations can be broken down into a four phase model: beliefs, subjective estimations, chance estimations and probabilistic calculations. The paper focuses on the development of the model over the course of the experiment, identifying the process the students underwent as they attempted to answer the question. We explain each phase as it was reflected in the students\'' rationalizations. All phases, including their definitions and students’ citations, will be presented in the paper. While not every student necessarily goes through all four phases, an awareness and understanding of them all allows for efficient, effective intervention during the learning process. We found that guidance and learning intervention helped shorten the preliminary phases, leading to more relative time spent on probabilistic calculations.