PROBABILITY AND CERTAINTY
... What we need now is a producer interpretation, or an input theory. The most general and simple input theory for probability is that the degree of belief that an event type E will have property P is equal to the ratio of observed events type E with property P to observed events of type E with or with ...
... What we need now is a producer interpretation, or an input theory. The most general and simple input theory for probability is that the degree of belief that an event type E will have property P is equal to the ratio of observed events type E with property P to observed events of type E with or with ...
Raymond J. Solomonoff 1926-2009 - Computer Science
... assigns a probability to the event that x will be continued by a sequence (or even just a symbol) y. In what follows we will have opportunity to appreciate the theoretical attractiveness of the formula: its prediction power, and its combination of a number of deep principles. But let us level with t ...
... assigns a probability to the event that x will be continued by a sequence (or even just a symbol) y. In what follows we will have opportunity to appreciate the theoretical attractiveness of the formula: its prediction power, and its combination of a number of deep principles. But let us level with t ...
3 - Rice University
... Let us play a small game. Bob is rolling a die. He asks Alice to guess what he rolled. 1. What is the probability that Alice is correct ? 2. Let us say that the Oracle told Alice that the outcome of the die is an even number or an odd number. Then what is the probability that Alice is correct ? ...
... Let us play a small game. Bob is rolling a die. He asks Alice to guess what he rolled. 1. What is the probability that Alice is correct ? 2. Let us say that the Oracle told Alice that the outcome of the die is an even number or an odd number. Then what is the probability that Alice is correct ? ...
Lesson 7: Infinite Decimals
... Show the first few stages of placing the decimal 0.777777… on the number line. ...
... Show the first few stages of placing the decimal 0.777777… on the number line. ...
An Introduction to Probability
... (this is where we can tell the state of both coins). Now we assume that neither coin knows the other’s intentions or outcome. This assumption restricts our choice of probability model quite considerably because it enforces a symmetry. Let us choose P ({hh, ht}) = p1h and P ({hh, th}) = p2h Now let u ...
... (this is where we can tell the state of both coins). Now we assume that neither coin knows the other’s intentions or outcome. This assumption restricts our choice of probability model quite considerably because it enforces a symmetry. Let us choose P ({hh, ht}) = p1h and P ({hh, th}) = p2h Now let u ...
Random numbers
... the sequence of random numbers is uncorrelated (i.e. numbers in the sequence are not related in any way). In numerical integration, it is important that the distribution is flat. 2. The sequence of random numbers must have a long period. All random number generators will repeat the same sequence of ...
... the sequence of random numbers is uncorrelated (i.e. numbers in the sequence are not related in any way). In numerical integration, it is important that the distribution is flat. 2. The sequence of random numbers must have a long period. All random number generators will repeat the same sequence of ...
Angles, Degrees, and Special Triangles
... – Do not need to worry about deriving the formula – Just know how to use it ...
... – Do not need to worry about deriving the formula – Just know how to use it ...
ON THE STRONG LAW OF LARGE NUMBERS FOR SEQUENCES
... 3.1. (i) The slower b; i 00, the stronger is the assumption (3.2), but so is the conclusion (3.3). (ii) Theorem 3.1 is an analogue of an SLLN of Adler et al. [2] obtained for a sequence of independent random elements in a Rademacher type p (1 ~ p ~ 2) Banach space. The Adler et al. [2] result, which ...
... 3.1. (i) The slower b; i 00, the stronger is the assumption (3.2), but so is the conclusion (3.3). (ii) Theorem 3.1 is an analogue of an SLLN of Adler et al. [2] obtained for a sequence of independent random elements in a Rademacher type p (1 ~ p ~ 2) Banach space. The Adler et al. [2] result, which ...
Axiomatic First-Order Probability
... probability 1 (0). This approach, natural as it seems, runs into difficulty. The first roadblock is that in standard first-order logic, arguments of functions must be elements of the domain, not sentences or propositions. The second roadblock is that the theory of the real numbers cannot be fully c ...
... probability 1 (0). This approach, natural as it seems, runs into difficulty. The first roadblock is that in standard first-order logic, arguments of functions must be elements of the domain, not sentences or propositions. The second roadblock is that the theory of the real numbers cannot be fully c ...
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.In this context, ""almost surely"" is a mathematical term with a precise meaning, and the ""monkey"" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the ""monkey metaphor"" is that of French mathematician Émile Borel in 1913, but the first instance may be even earlier. The relevance of the theorem is questionable—the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). It should also be noted that real monkeys don't produce uniformly random output, which means that an actual monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text.Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.