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Topic: Law of Total Probability

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Bayes factor

	Law of total expectation - Wikipedia, the free encyclopedia
		The proposition in probability theory known as the law of total expectation, or the law of iterated expectations, or perhaps by any of a variety of other names, states that if X is an integrable random variable (i.e., a random variable satisfying E(
		The nomenclature used here parallels the phrase law of total probability.
		Notice that the conditional expected value of X given the event Y = y is a function of y (this is where adherence to the conventional rigidly case-sensitive notation of probability theory becomes important!).
	en.wikipedia.org /wiki/Law_of_total_expectation (208 words)

	Statistics Glossary - probability
		Probability is conventionally expressed on a scale from 0 to 1; a rare event has a probability close to 0, a very common event has a probability close to 1.
		Like all probabilities, a subjective probability is conventionally expressed on a scale from 0 to 1; a rare event has a subjective probability close to 0, a very common event has a subjective probability close to 1.
		The addition rule is a result used to determine the probability that event A or event B occurs or both occur.
	www.stats.gla.ac.uk /steps/glossary/probability.html (1632 words)

	Law of total probability (Site not responding. Last check: 2007-09-10)
		Nomenclature in probability theory is not wholly standard.
		The phrase law of total probability is also used to refer to proposition that says that under similar assumptions have
		The prior probability of A is equal to the prior expected value of the posterior probability of A.
	www.freeglossary.com /Total_probability (346 words)

	[No title]
		The probability of future occurrences of an event can be solved using this formula with an event E that occurred p times and failed q times; p and q are integers.
		The earlier event is known as the prior probability and what one is solving for, the recent event given the earlier has occurred, is called the posterior probability (Ghahramani 101).
		If one wanted to find the probability of an American man’s height is over one and one half meters, the result would depend if he or she knows anything about the man. If no information is known the probability is equal to the proportion of American men taller than the specified height.
	www.saintjoe.edu /~karend/m441/KarenFinalPaper.doc (2860 words)

	Applications of conditional probability (from probability theory) -- Britannica Student Encyclopedia
		An application of the law of total probability to a problem originally posed by Christiaan Huygens is to find the probability of “gambler's ruin.” Suppose two players, often called Peter and Paul, initially have x and m
		Probability has its origin in the study of gambling and insurance in the 17th century, and it is now an indispensable tool of both social and natural sciences.
		In genetics, for example, probability is used to estimate the likelihood for brown-eyed parents to produce a blue-eyed child (see Heredity).
	www.britannica.com /ebi/article-32768 (865 words)

	Conditional Probability
		The denominator of this quantity is the probability of receiving a morning paper (being in the subset).
		The numerator is the probability of being in both the restricted subset AND having the condition specified.
		We compute a ratio where the denominator is the probability of being in the given subset B, and the numerator is the probability of being in both sets.
	www.ms.uky.edu /~viele/sta531f00/condprob/condprob.html (1736 words)

	Math 5710 - Introduction to Probability (Site not responding. Last check: 2007-09-10)
		Probability, the study of random or chance processes, is widely used in many fields of study.
		Random Variables: discrete random variables; cumulative distribution functions; probability mass functions; bernoulli, binomial, negative binomial, hypergeometric, and Poisson distributions; continuous random variables; probability density functions; uniform, exponential, gamma, normal, and beta distributions; distributions of functions of random variables; expectation, mean, and variance; Markov's and Chebyshev's inequalities; moment-generating, characteristic, and probability-generating functions.
		B) In a game of poker, show that the probabilities that a five-card hand will contain (a) a straight (five cards in numerical order), (b) four of a kind, and (c) a full house (three cards of one rank, two of another) are (a).0035, (b).00024, and (c).0014, respectively.
	www.math.usu.edu /~minnotte/M5710Su00 (966 words)

	Statistics Glossary - Probability
		In probability theory we say that two events, A and B, are independent if the probability that they both occur is equal to the product of the probabilities of the two individual events.
		The probability distribution of a discrete random variable is a list of probabilities associated with each of its possible values.
		The probability density function of a continuous random variable is a function which can be integrated to obtain the probability that the random variable takes a value in a given interval.
	www.cas.lancs.ac.uk /glossary_v1.1/prob.html (3540 words)

	Statistics:Probability - Wikibooks, collection of open-content textbooks
		A basic understanding of probability is vital in grasping basic statistics, and probability is largely abstract without statistics to determine the "real world" probabilities.
		This section is not meant to give a comprehensive lecture in probability, but rather simply touch on the basics that are needed for this class, covering the basics of Bayesian Analysis for those students who are looking for something a little more interesting.
		Also for events A do we speak of their probability P(A) (written with a capital P), which is simply the total of the probabilities of the outcomes in A. For a fair dice p(s) = 1/6 for each outcome s and P("even") = P(E) = 1/6+1/6+1/6 = 1/2.
	en.wikibooks.org /wiki/Statistics:Probability (686 words)

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		0, the conditional probability of A given B, denoted by P(AB), is EMBED Equation.3 Theorem 3.3.
		Law of Total Probability Let B be an event with P(B)>0 and P(Bc)>0.
		An electric circuit has four switches that are independently closed or open with probabilities p and 1-p respectively.
	www.ams.sunysb.edu /~finchs/AMS311Lecture6.doc (572 words)

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		Conditional probability for events is the probability that event E will happen, given that event F has already occurred.
		Law of Total Probability involves applying the probability by conditioning and adding all the probabilities.
		Taking conditional probabilities of various events with respect to a given event F amounts to choosing F as a new sample space; and we have to multiply all probabilities by 1/ Pr[F] in order to reduce the total probability to 1.
	www.math.unl.edu /~sdunbar1/Teaching/MathematicalFinance/Lessons/Conditionals/ConditionalProbability/condprob.shtml (1741 words)

	Solving Problem 1: The Evaluation Problem
		One method is to use the law of total probability and condition on all possible state sequences, I, of length T:
		The standard approach is to use the forward-backward procedure, which takes advantage of the Markov property, namely that the path to a state is irrelevant to the future behavior of the process after being in that state.
		which is the probability of the process producing the first t observations and of ending up in state j at time t.
	www.cs.tufts.edu /g/232/hmm/node2.html (235 words)

	Law of total variance (Site not responding. Last check: 2007-09-10)
		In probability theory the law of total variance states that if X and Y are random variables on the same probability space and the variance of X is finite then
		Notice that the conditional expected value X given the event Y = y is a function of y (this is where adherence to the rigidly case-sensitive notation of probability theory becomes If we write E(y) = g (y) then the random variable E() is just g (Y).
		Generalizations for higher moments than the third messy; for higher cumulants on the other hand a simple elegant form exists.
	www.freeglossary.com /Law_of_total_variance (325 words)

	[No title]
		Law of Multiplication: EMBED Equation.3 Remember that we used this in the Let’s Make a Deal example.
		Generalized Law of Total Probabililty If EMBED Equation.3 is a partition of the sample space of an experiment and P(Bi)>0 for EMBED Equation.3 , then for any event A of S, EMBED Equation.3 Application to overall mortality rate; definition of directly standardized rate.
		Gambler’s Ruin Problem Two gamblers play the game of “heads or tails,” in which each time a fair coin lands heads up, player A wins $1 from B, and each time it lands tails up, player B wins $1 from A. Suppose that player A initially has a dollears and player B has b dollars.
	www.ams.sunysb.edu /~finchs/lec6.doc (473 words)

	Lecture Menu
		The event-composition approach to calculate the probability of an event of interest, say event A, expresses A as a composition (unions and/or intersections) of two or more other events.
		Then the two laws of probability can be applied to the composition to find P(A).
		It knows from previous experience that if oil is to be found, there is a probability of 0.4 that a positive strike of some kind will be made on the first series of drillings.
	www.math.sfu.ca /~lidels/stat270/lecture7&8.htm (798 words)

	No Title
		Since it is mathematically inconvenient to keep redefining the sample space, we simply define a new probability measure which will assign probability 0 to sets that do not not meet the requirements of the partial information.
		To do this, we want to assign probability 0 to all events which do not intersect the event that the first card is the queen of hearts.
		This is called the law of total probability, and is simply another way of formulating our ``divide and conquer'' for computing probabilities.
	www.uwm.edu /~ericskey/361material/361F98/L05/L05.html (606 words)

	[No title] (Site not responding. Last check: 2007-09-10)
		Two bolts are randomly selected from the total production the probability that they are both out of spec is: (a) 0.1 (b) 0.059 (c) 0.00348 (d) 0.00696 (e) 0.118 5.
		Two bolts are randomly selected the probability that they were both produced by A is: (a) 0.16 (b) 0.01 (c) 0.04 (d) 0.8 (e) 0.21 6.
		Two bolts are selected it is known that they were both produced by the same machine the probability that exactly one is out of spec is: (a) 0.1209 (b) 0.1087 (c) 0.0923 (d) 0.1823 (e) 0.16 8.
	www.maths.tcd.ie /~donalmcc/DMCC/3E2/Tutorials/Tutorial02_04.doc (526 words)

	Advanced Probability Theory at the University of Zimbabwe (Site not responding. Last check: 2007-09-10)
		Probability: Review of set theory, infinite collections of sets, sigma fields, Borel sets, events, probability space, Kolmogorov axioms, conditional probability, law of total probability, continuity of probability measure, independence, Borel-Cantelli lemmas.
		Probability distributions: Random variables, univariate and multivariate probability distributions, Kolmogorov extension theorem, independence, Zero One law, expectation, characteristic functions.
		Weak and strong laws of large numbers, law of iterated logarithm, central limit theorem.
	www.uz.ac.zw /science/maths/courses/hmth342.htm (149 words)

	[No title]
		The cdf for a random variable gives the probability that the RV will take on a value less than or equal to some possible value.
		The idea of the method of maximum likelihood is to find the value of $\theta$ that maximizes $L$, and use that value as an estimator/estimate of $\theta$.
		In effect, we're asking the question, ``for what value of $\theta$ would the data be most probable?'' Often, it will be much easier to maximize the natural log of the likelihood function rather than the likelihood function itself.
	www.sscnet.ucla.edu /ssc/labs/brenner/econ143/prob.html (1315 words)

	MIT OpenCourseWare \| Mathematics \| 18.441 Statistical Inference, Spring 2002 \| Lecture Notes (Site not responding. Last check: 2007-09-10)
		We began with a fairly typical maximum-likelihood problem, and found that the maximum-likelihood estimator in that case is unbiased (and in passing, we defined the concepts of "biased" and "unbiased").
		The total corrected sum of squares measures the variability in the data; the separate rows in the ANOVA table correspond to separate "sources" of variability.
		Today we looked at the Kolmogorov-Smirnov test, which relies on the "maximum-discrepancy statistic." The fact that the probability distribution of the test statistic given that the null hypothesis is true, does not depend on which continuous distribution the null hypothesis specifies, may be surprising initially.
	ocw.mit.edu /OcwWeb/Mathematics/18-441Statistical-InferenceSpring2002/LectureNotes (3067 words)

	Conditional Probability and Independent Events (Site not responding. Last check: 2007-09-10)
		What is the conditional probability that their children are both boys, given that they have at least one son?
		Suppose that we assume that 30 percent of all the new policyholders of an insurance company is accident prone.
		Then what is the probability that a new policyholder will have an accident within a year of purchasing a policy?
	math.tntech.edu /machida/3470/booklet/booklet/node4.html (182 words)

	Homework on Data Augmentation
		Recall the law of large numbers method for approximating an expectation.
		Item 3 is a law of large numbers approximation.
		You should be able to adapt many of the formulas we discussed in class for the EM algorithm to the Data augmentation algorithm.
	www.ms.uky.edu /~viele/sta695s99/homeworkda/homeworkda.html (693 words)

	AS 307 Section 001 (Site not responding. Last check: 2007-09-10)
		The customer will purchase both suit and a shirt with probability 0.11, both suit and a tie with probability 0.14, and both shirt and a tie with probability 0.10.
		A man staying at a parisian hotel writes this word, and a letter taken at random from his spelling is found to be vowel.
		What is the probability that a person selected at random who contributed to one of the batches was virus carrier?
	hosting.uaa.alaska.edu /afkt/as307gu1.html (415 words)

	Conditional Probability and Related Topics (Site not responding. Last check: 2007-09-10)
		The estimate P(A) of the probability of an event A, may change if we are given the fact that some other event B has occurred.
		The conditional probability of A, given that B has occurred is defined as
		Suppose A and B are two events having positive probabilities.
	www.pitt.edu /~jrclass/e20/notes/OH5.html (132 words)

	April 3, 2000
		A hiker leaves the point O shown in Figure 1, choosing one of the roads OB1, OB2, OB3 or OB4 at random.
		What is the probability that a red ball is selected from the first box AND a red ball is selected from the second box?
		At the conclusion of the selection process, what is the probability that the numbers of red and green balls in the firs box are identical to the numbers at the beginning?
	www.mcs.drexel.edu /~omokliat/courses/mcs311F00/HANDOUTS/HAND11.html (256 words)

	Probability & Statistics - Course Profile (Site not responding. Last check: 2007-09-10)
		We will study probability theory, including conditional probability, the Law of Total Probability and Bayes' Law, random variables and their distributions, laws of large numbers and the Central Limit Law, Markov chains, and basic statistical models.
		Frequencies in repeated experiments, the probability axioms, simple properties of probability measure, finite experiments, the equilikely principle.
		Convergence of random variables, Chebyshev's inequality, convergence in probability, proof of the Weak Law of Large Numbers, statement of the Strong Law of Large Numbers and illustrations, convergence in mean and in mean square, convergence in distribution, the Central Limit Theorem.
	www.maths.uq.edu.au /courses/STAT2003/05profile.html (2570 words)

	STAT2003 Probability & Statistics - Course Profile
		We will study probability theory, including conditional probability, the Law of Total Probability and Bayes' Law, random variables and their distributions, laws of large numbers and the Central Limit Law, Markov processes, and basic statistical models.
		Convergence of RVs, Chebyshev's inequality, convergence in probability, proof of the Weak Law of Large Numbers, statement of the Strong Law of Large Numbers and illustrations, convergence in mean and in mean square, convergence in distribution, the Central Limit Theorem
		This will be based on 5 assignments each worth 6%, making up a total of 30% of the final mark, and an examination making up the remaining 70%.
	www.maths.uq.edu.au /courses/Profiles/2003/stat2003_s1_03.htm?MySourceSession=9dbd12b1d50f3ec3d34e8dabef583aed (2580 words)

	MAT 370 Probability (Site not responding. Last check: 2007-09-10)
		Three primary goals are these: a) to understand the pervasive presence of randomness in the everyday world, nature, and in the phenomena investigated by science; b) to understand the general rules that govern random processes; c) to learn how to create mathematical models of random processes.
		Law of Total Probability; basic rules of conditional probability; independence; Bayes' formula
		probability distributions; bernoulli, geometric, and binomial distributions; expectations; sampling with and without replacement
	www.warren-wilson.edu /~teller/dept/mat_370_syl.htm (585 words)

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