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6.16 independent and dependent practice
... spins a spinner with the letters M, O, N, E, and Y. What is the probability that the spinner will stop on the letter N on the first spin and an O the second spin? ...
... spins a spinner with the letters M, O, N, E, and Y. What is the probability that the spinner will stop on the letter N on the first spin and an O the second spin? ...
UNCERTAINTY THEORIES: A UNIFIED VIEW
... • Generalized p-boxes are special random sets that generalize BOTH p-boxes and possibility distributions • Clouds extend GP-boxes but induce lower probabilities that are not even 2-monotonic. • Probability intervals are not comparable to generalized p-boxes: they induce lower probabilities that are ...
... • Generalized p-boxes are special random sets that generalize BOTH p-boxes and possibility distributions • Clouds extend GP-boxes but induce lower probabilities that are not even 2-monotonic. • Probability intervals are not comparable to generalized p-boxes: they induce lower probabilities that are ...
Lecture 3 Uncertainty management in rule- based
... goes back thousands of years when words like “probably”, “likely”, “maybe”, “perhaps” and “possibly” were introduced into spoken languages. However, the mathematical theory of probability was formulated only in the 17th century. I The probability of an event is the proportion of cases in which the e ...
... goes back thousands of years when words like “probably”, “likely”, “maybe”, “perhaps” and “possibly” were introduced into spoken languages. However, the mathematical theory of probability was formulated only in the 17th century. I The probability of an event is the proportion of cases in which the e ...
Bayes` Rule
... in a factory. From the batch allotted to them Aileen packs 55%, Barbara 30% and Cathy 15%. The probability that Aileen breaks some biscuits in a packet is 0.7 and the respective probabilities for Barbara and Cathy are 0.2 and 0.1. What is the probability that a packet with broken biscuits found by t ...
... in a factory. From the batch allotted to them Aileen packs 55%, Barbara 30% and Cathy 15%. The probability that Aileen breaks some biscuits in a packet is 0.7 and the respective probabilities for Barbara and Cathy are 0.2 and 0.1. What is the probability that a packet with broken biscuits found by t ...
Early Work – Oct. 16
... At this rate, what would a 25-pound bag cost? Washington apples are selling for 50 cents a pound. If on the average, 2 apples equal one pound, what would be the cost of 20 apples? ...
... At this rate, what would a 25-pound bag cost? Washington apples are selling for 50 cents a pound. If on the average, 2 apples equal one pound, what would be the cost of 20 apples? ...
Foundations of Reasoning 1 Logic
... Bayes Theorem: p(E|F ) = p(F |E) p(E) p(F ) Bayes theorem is important because it expresses the quantity p(E|F ) (the probability of a hypothesis E given the evidence F ) — which is something people often find hard to assess — in terms of quantities that can be drawn directly from experiential knowl ...
... Bayes Theorem: p(E|F ) = p(F |E) p(E) p(F ) Bayes theorem is important because it expresses the quantity p(E|F ) (the probability of a hypothesis E given the evidence F ) — which is something people often find hard to assess — in terms of quantities that can be drawn directly from experiential knowl ...
Presentation (PowerPoint File)
... Bayesian view: assess expected amount of information from each card (cf Lindley 1956) ...
... Bayesian view: assess expected amount of information from each card (cf Lindley 1956) ...
Review
... c) What is the theoretical probability of receiving the following? • P(Red, Green) ________________ • P(Yellow, Yellow) ____________ • P(Blue, Red) _________________ • P(Yellow, Green) _____________ 3. Conduct an experiment of 10, 20, and 30 trials to simulate question 2. When was the experimental p ...
... c) What is the theoretical probability of receiving the following? • P(Red, Green) ________________ • P(Yellow, Yellow) ____________ • P(Blue, Red) _________________ • P(Yellow, Green) _____________ 3. Conduct an experiment of 10, 20, and 30 trials to simulate question 2. When was the experimental p ...
Statistical Inference
... hypotheses because: 1) hypotheses should be compared by how well they explain the data. 2) the p-value does not account for how well the alternative hypotheses explain the data 3) the p-value summands are irrelevant because they don’t explain how well any hypothesis explains any observed data. In sh ...
... hypotheses because: 1) hypotheses should be compared by how well they explain the data. 2) the p-value does not account for how well the alternative hypotheses explain the data 3) the p-value summands are irrelevant because they don’t explain how well any hypothesis explains any observed data. In sh ...
PROBABILITY IS SYMMETRY
... (i) Data: Boys are born with frequency 0.513 (ii) Law of Large Numbers Subjective interpretation of probability (i) Use mathematical probability to express uncertainty (ii) Given new information (data), update your opinion using the Bayes Theorem (iii) Make decisions that maximize the expected gain ...
... (i) Data: Boys are born with frequency 0.513 (ii) Law of Large Numbers Subjective interpretation of probability (i) Use mathematical probability to express uncertainty (ii) Given new information (data), update your opinion using the Bayes Theorem (iii) Make decisions that maximize the expected gain ...
Lecture 1. Probabilities - Definitions, Examples and Basic Tools
... about outcome of an experiment, that express the chances that the statement is true. Statistics is the scientific application of mathematical principles to the collection, analysis, and presentation of ...
... about outcome of an experiment, that express the chances that the statement is true. Statistics is the scientific application of mathematical principles to the collection, analysis, and presentation of ...
6.3 Calculator Examples
... • Our calculator can also directly calculate binomial probabilities • Binompdf(n,p,k) computes the probability that X=k • Binomcdf(n,p,k) computes the probability that X≤k – Remember, n is the number of trials – P is the probability of success in any given trial ...
... • Our calculator can also directly calculate binomial probabilities • Binompdf(n,p,k) computes the probability that X=k • Binomcdf(n,p,k) computes the probability that X≤k – Remember, n is the number of trials – P is the probability of success in any given trial ...
Data analysis: Frequently Bayesian
... An important problem is that specification of ignorance for a continuous parameter is not unique. For example, a model may be parameterized not by θ but instead by λ = ln θ. A constant probability for one parameter would imply a nonconstant probability for the other. Nevertheless, one often uses uni ...
... An important problem is that specification of ignorance for a continuous parameter is not unique. For example, a model may be parameterized not by θ but instead by λ = ln θ. A constant probability for one parameter would imply a nonconstant probability for the other. Nevertheless, one often uses uni ...
Bayesianism with a Human Face - Minnesota Center for Philosophy
... beliefs must be represented by a unique probability. But de Finetti was far from saying that personal probabilities cannot fail to exist. (It is a separate question, whether one can be unaware of one's existent partial beliefs. I don't see why not. See Mellor (1980) and Skyrms (1980) for extensive d ...
... beliefs must be represented by a unique probability. But de Finetti was far from saying that personal probabilities cannot fail to exist. (It is a separate question, whether one can be unaware of one's existent partial beliefs. I don't see why not. See Mellor (1980) and Skyrms (1980) for extensive d ...
Conditional probability and independence Bernoulli trials and the
... P (Ak |B) = and the law of total probability. ...
... P (Ak |B) = and the law of total probability. ...
Probability Unit
... A game consists of rolling a colored die with three red sides, two green sides, and one blue side. A roll of red loses. A role of green pays $2.00. A roll of blue pays $5.00. The charge to play the game is $2.00. Would you play the game? Why or why not? ...
... A game consists of rolling a colored die with three red sides, two green sides, and one blue side. A roll of red loses. A role of green pays $2.00. A roll of blue pays $5.00. The charge to play the game is $2.00. Would you play the game? Why or why not? ...
Chapter 4: Probability Rare Event Rule for Inferential Statistics Rare
... many times, the relative frequency probability (from Rule 1) of an event tends to approach the actual probability (Rule 2 for equally likely outcomes) ...
... many times, the relative frequency probability (from Rule 1) of an event tends to approach the actual probability (Rule 2 for equally likely outcomes) ...
Handling Uncertainties - using Probability Theory to
... independent of each other and this helps to resolve the uncertainty. This relationship has the following mathematical property : P( A, B) = P( A).P( B) ...
... independent of each other and this helps to resolve the uncertainty. This relationship has the following mathematical property : P( A, B) = P( A).P( B) ...
probability rules
... P(A ∩ B) = P(A) P(B | A) = P(B) P(A | B) Using your estimates from the previousl page and the multiplication rule above, compute the probability that: The red die is 2 and green die is 3 A randomly chosen Calvin student is a male engineer Both the Packers and the Forty-Niners win Sunday The top two ...
... P(A ∩ B) = P(A) P(B | A) = P(B) P(A | B) Using your estimates from the previousl page and the multiplication rule above, compute the probability that: The red die is 2 and green die is 3 A randomly chosen Calvin student is a male engineer Both the Packers and the Forty-Niners win Sunday The top two ...
Probability of Independent Events
... • Independent event - two events whose occurrence of one event DOES NOT affect the likelihood that the other event will occur • Examples: – Fliping a coin and spinning a spinner – drawling a marble, replacing it, and then drawling another marble – a girl puppy is born first and a boy puppy is born s ...
... • Independent event - two events whose occurrence of one event DOES NOT affect the likelihood that the other event will occur • Examples: – Fliping a coin and spinning a spinner – drawling a marble, replacing it, and then drawling another marble – a girl puppy is born first and a boy puppy is born s ...
on p (D+ YA+)
... The likelihood is the probability of the data given the parameter and represents the data now available. The posterior represents what is thought given both prior information and the data just seen. It relates the conditional density of a parameter (posterior probability) with its unconditional dens ...
... The likelihood is the probability of the data given the parameter and represents the data now available. The posterior represents what is thought given both prior information and the data just seen. It relates the conditional density of a parameter (posterior probability) with its unconditional dens ...
Lecture 1. Probabilities - Definitions, Examples
... Term experiment is used to refer to any process whose outcome is not known in advance. Consider an experiment. I ...
... Term experiment is used to refer to any process whose outcome is not known in advance. Consider an experiment. I ...
... candidates for determining reference classes for hypothesessimplicity, for example, seem likely to give perverse results. We prefer hypotheses that posit simple relations among observed quantities, and so on a frequentist view should give them high prior probabilities. Yet simple hypotheses, althoug ...
Dempster–Shafer theory
The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty, with understood connections to other frameworks such as probability, possibility and imprecise probability theories. First introduced by Arthur P. Dempster in the context of statistical inference, the theory was later developed by Glenn Shafer into a general framework for modeling epistemic uncertainty - a mathematical theory of evidence. The theory allows one to combine evidence from different sources and arrive at a degree of belief (represented by a mathematical object called belief function) that takes into account all the available evidence.In a narrow sense, the term Dempster–Shafer theory refers to the original conception of the theory by Dempster and Shafer. However, it is more common to use the term in the wider sense of the same general approach, as adapted to specific kinds of situations. In particular, many authors have proposed different rules for combining evidence, often with a view to handling conflicts in evidence better. The early contributions have also been the starting points of many important developments, including the Transferable Belief Model and the Theory of Hints.