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Chapter 3: Random Graphs 3.1 G(n,p) model ( )1 Chapter 3
... graphs from traditional graph theory and graph algorithms is that here one seeks statistical properties of these very large graphs rather than an exact answer to questions. This is perhaps akin to the switch Physics made in the late 19th century in going from Mechanics to Statistical Mechanics. Just ...
... graphs from traditional graph theory and graph algorithms is that here one seeks statistical properties of these very large graphs rather than an exact answer to questions. This is perhaps akin to the switch Physics made in the late 19th century in going from Mechanics to Statistical Mechanics. Just ...
probability - Jobpulp.com
... Example 9 : Harpreet tosses two different coins simultaneously (say, one is of Re 1 and other of Rs 2). What is the probability that she gets at least one head? Solution : We write H for ‘head’ and T for ‘tail’. When two coins are tossed simultaneously, the possible outcomes are (H, H), (H, T), (T, ...
... Example 9 : Harpreet tosses two different coins simultaneously (say, one is of Re 1 and other of Rs 2). What is the probability that she gets at least one head? Solution : We write H for ‘head’ and T for ‘tail’. When two coins are tossed simultaneously, the possible outcomes are (H, H), (H, T), (T, ...
RANDOM NUMBERS, MONTE CARLO METHODS
... WHAT IS A RANDOM NUMBER? Is 2.71947 a random number? Not at all, but it was (seriously!) randomly generated (by me). It’s easier to meaningfully talk about a random set or sequence of numbers following a particular probability distribution and about random generation of numbers. • The precise math ...
... WHAT IS A RANDOM NUMBER? Is 2.71947 a random number? Not at all, but it was (seriously!) randomly generated (by me). It’s easier to meaningfully talk about a random set or sequence of numbers following a particular probability distribution and about random generation of numbers. • The precise math ...
the BIRTHDAY problem
... people share the same birthday. In a room with 60 or more people the probability that at least 2 people share the same birthday is greater than 99%. It is a paradox in the sense that it is a mathematical truth that contradicts common intuition (most people estimate the chance of having 2 people with ...
... people share the same birthday. In a room with 60 or more people the probability that at least 2 people share the same birthday is greater than 99%. It is a paradox in the sense that it is a mathematical truth that contradicts common intuition (most people estimate the chance of having 2 people with ...
Grade 10 Probability
... would be), but just the total possible arrangement Step 2 T he answer to this is that there are 11 letters in this word, and theref ore the letters can be arranged in 11! ways i.e. 11x10x9x...2x1 ways T o see this, take any of the letters - it can be in any of the possible 11 positions. For each of ...
... would be), but just the total possible arrangement Step 2 T he answer to this is that there are 11 letters in this word, and theref ore the letters can be arranged in 11! ways i.e. 11x10x9x...2x1 ways T o see this, take any of the letters - it can be in any of the possible 11 positions. For each of ...
On a strong law of large numbers for monotone measures
... We also denote VµC (X) = EµC (X − EµC [X])2 . Throughout this paper, we always consider the existence of EµC (X) and of VµC (X). Notice that if µ is probability measure, µ = P , then EµC (X) = EPC (X) = E (X) and VµC (X) = VPC (X) = var (X) . In order to consider the convergence analysis of random v ...
... We also denote VµC (X) = EµC (X − EµC [X])2 . Throughout this paper, we always consider the existence of EµC (X) and of VµC (X). Notice that if µ is probability measure, µ = P , then EµC (X) = EPC (X) = E (X) and VµC (X) = VPC (X) = var (X) . In order to consider the convergence analysis of random v ...
Interpreting Probability - Assets - Cambridge
... to notice that inverting the equations changed the meaning of probability. The probability of picking a ball of a particular color was a function of the relative proportions in the urn. The probability of guilt was different: it was a quantified judgment in the light of a specified body of knowledge ...
... to notice that inverting the equations changed the meaning of probability. The probability of picking a ball of a particular color was a function of the relative proportions in the urn. The probability of guilt was different: it was a quantified judgment in the light of a specified body of knowledge ...
PROBABILITY THEORY - PART 2 INDEPENDENT RANDOM
... answers, we shall weave through various topics. Here is a guide to the essential aspects that you might pay attention to. Firstly, the results. We shall cover fundamental limit theorems of probability, such as the weak and strong law of large numbers, central limit theorems, poisson limit theorem, i ...
... answers, we shall weave through various topics. Here is a guide to the essential aspects that you might pay attention to. Firstly, the results. We shall cover fundamental limit theorems of probability, such as the weak and strong law of large numbers, central limit theorems, poisson limit theorem, i ...
The Parity of Set Systems under Random Restrictions
... probability. When compared to previously known reductions of this type, ours excel in their simplicity: For graph problems, restricting elements of the ground set usually corresponds to simple deletion and contraction operations, which can be encoded efficiently in most problems. We find three appli ...
... probability. When compared to previously known reductions of this type, ours excel in their simplicity: For graph problems, restricting elements of the ground set usually corresponds to simple deletion and contraction operations, which can be encoded efficiently in most problems. We find three appli ...
Problem Set 3 [Word]
... different colors—yellow, green, blue, and magenta, respectively. Suppose I apply a Vigenère cipher to this plaintext. Depending on the length of the keyword I use, it’s possible that some of these woods would be enciphered identically. For each of the following keyword lengths, determine which woods ...
... different colors—yellow, green, blue, and magenta, respectively. Suppose I apply a Vigenère cipher to this plaintext. Depending on the length of the keyword I use, it’s possible that some of these woods would be enciphered identically. For each of the following keyword lengths, determine which woods ...
Infinite monkey theorem
![](https://commons.wikimedia.org/wiki/Special:FilePath/Monkey-typing.jpg?width=300)
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.