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Lottery Mathematics
Lottery mathematics is used to calculate probabilities of winning or losing a lottery game. It is based primarily on combinatorics, particularly the twelvefold way and combinations without replacement. It can also be used to analyze coincidences that happen in lottery drawings, such as repeated numbers appearing across different draws.M. Pollanen (2024) ''A Double Birthday Paradox in the Study of Coincidences'', Mathematics 23(24), 3882. https://doi.org/10.3390/math12243882 Choosing 6 from 49 In a typical 6/49 game, each player chooses six distinct numbers from a range of 1–49. If the six numbers on a ticket match the numbers drawn by the lottery, the ticket holder is a jackpot winner— regardless of the order of the numbers. The probability of this happening is 1 in 13,983,816. The chance of winning can be demonstrated as follows: The first number drawn has a 1 in 49 chance of matching. When the draw comes to the second number, there are now only 48 balls left in the bag, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilities
Probability is a branch of mathematics and statistics concerning Event (probability theory), events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th ed., (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', vol. 1, 3rd ed., (1968), Wiley, . This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an Probab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lotto 6/49
Lotto 6/49 is one of three national lottery games in Canada. Launched on June 12, 1982, Lotto 6/49 was the first nationwide Canadian lottery game to allow players to choose their own numbers. Previous national games, such as the Olympic Lottery, Loto Canada and Superloto used pre-printed numbers on tickets. Lotto 6/49 led to the gradual phase-out of that type of lottery game in Canada. Winning numbers are drawn by the Interprovincial Lottery Corporation (ILC) every Wednesday and Saturday, executed with a random number generator. Gameplay As of the September 14, 2022 draw, the game consists of two components: * The "Gold Ball Draw", formerly the "guaranteed prize draw", which is a raffle prize of at least $1 million that is awarded during each draw. Some drawings (promoted as a "Superdraw") offer multiple secondary raffle prizes. * The "Classic Draw", in which six numbers are drawn from a set of 49. If a ticket matches all six numbers, a fixed prize of CA$5 million is won. A bonu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Distribution
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with probability q = 1-p. Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcome (probability), outcomes that are Boolean-valued function, Boolean-valued: a single bit whose value is success/yes and no, yes/Truth value, true/Binary code, one with probability ''p'' and failure/no/false (logic), false/Binary code, zero with probability ''q''. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and ''p'' would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and ''p'' would be the probability of tails). In particular, unfair co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which * the domain is the set of possible outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a die; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Content
In information theory, the information content, self-information, surprisal, or Shannon information is a basic quantity derived from the probability of a particular event occurring from a random variable. It can be thought of as an alternative way of expressing probability, much like odds or log-odds, but which has particular mathematical advantages in the setting of information theory. The Shannon information can be interpreted as quantifying the level of "surprise" of a particular outcome. As it is such a basic quantity, it also appears in several other settings, such as the length of a message needed to transmit the event given an optimal source coding of the random variable. The Shannon information is closely related to ''entropy'', which is the expected value of the self-information of a random variable, quantifying how surprising the random variable is "on average". This is the average amount of self-information an observer would expect to gain about a random variable whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the mathematical study of the quantification (science), quantification, Data storage, storage, and telecommunications, communication of information. The field was established and formalized by Claude Shannon in the 1940s, though early contributions were made in the 1920s through the works of Harry Nyquist and Ralph Hartley. It is at the intersection of electronic engineering, mathematics, statistics, computer science, Neuroscience, neurobiology, physics, and electrical engineering. A key measure in information theory is information entropy, entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a Fair coin, fair coin flip (which has two equally likely outcomes) provides less information (lower entropy, less uncertainty) than identifying the outcome from a roll of a dice, die (which has six equally likely outcomes). Some other important measu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Axioms
The standard probability axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. There are several other (equivalent) approaches to formalising probability. Bayesians will often motivate the Kolmogorov axioms by invoking Cox's theorem or the Dutch book arguments instead. Kolmogorov axioms The assumptions as to setting up the axioms can be summarised as follows: Let (\Omega, F, P) be a measure space such that P(E) is the probability of some event E, and P(\Omega) = 1. Then (\Omega, F, P) is a probability space, with sample space \Omega, event space F and probability measure P. First axiom The probability of an event is a non-negative real number: :P(E)\in\mathbb, P(E)\geq 0 \qquad \forall E \in F where F is the event space. It follows (when combined with the second axiom) that P(E) is alwa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Event (probability Theory)
In probability theory, an event is a subset of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event consisting of only a single outcome is called an or an ; that is, it is a singleton set. An event that has more than one possible outcome is called a compound event. An event S is said to if S contains the outcome x of the experiment (or trial) (that is, if x \in S). The probability (with respect to some probability measure) that an event S occurs is the probability that S contains the outcome x of an experiment (that is, it is the probability that x \in S). An event defines a complementary event, namely the complementary set (the event occurring), and together these define a Bernoulli trial: did the event occur or not? Typically, when the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atom (measure Theory)
In mathematics, more precisely in measure theory, an atom is a measurable set that has positive measure and contains no set of smaller positive measures. A measure that has no atoms is called non-atomic or atomless. Definition Given a measurable space (X, \Sigma) and a measure \mu on that space, a set A\subset X in \Sigma is called an atom if \mu(A) > 0 and for any measurable subset B \subseteq A, either \mu(B) = 0 or \mu(B)=\mu(A). The equivalence class of A is defined by := \, where \Delta is the symmetric difference operator. If A is an atom then all the subsets in /math> are atoms and /math> is called an atomic class. If \mu is a \sigma-finite measure, there are countably many atomic classes. Examples * Consider the set ''X'' = and let the sigma-algebra \Sigma be the power set of ''X''. Define the measure \mu of a set to be its cardinality, that is, the number of elements in the set. Then, each of the singletons , for ''i'' = 1, 2, ..., 9, 10 is an atom. * Consider ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outcome (probability)
In probability theory, an outcome is a possible result of an Experiment (probability theory), experiment or trial. Each possible outcome of a particular experiment is unique, and different outcomes are Mutually exclusive events, mutually exclusive (only one outcome will occur on each trial of the experiment). All of the possible outcomes of an experiment form the elements of a sample space. For the experiment where we flip a coin twice, the four possible ''outcomes'' that make up our ''sample space'' are (H, T), (T, H), (T, T) and (H, H), where "H" represents a "heads", and "T" represents a "tails". Outcomes should not be confused with Event (probability theory), events, which are (or informally, "groups") of outcomes. For comparison, we could define an event to occur when "at least one 'heads'" is flipped in the experiment - that is, when the outcome contains at least one 'heads'. This event would contain all outcomes in the sample space except the element (T, T). Sets of outco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a . A probability space consists of three elements:Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press. # A '' sample space'', \Omega, which is the set of all possible outcomes of a random process under consideration. # An event space, \mathcal, which is a set of events, where an event is a subset of outcomes in the sample space. # A '' probability function'', P, which assigns, to each event in the event space, a probability, which is a number between 0 and 1 (inclusive). In order to provide a model of probability, these elements must satisfy probability axioms. In the example of the throw of a standard die, # The sample space \Omega is typically the set \ where each element in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Measure
In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if it is concentrated on an at most countable set. The support need not be a discrete set. Geometrically, a discrete measure (on the real line, with respect to Lebesgue measure) is a collection of point masses. Definition and properties Given two (positive) σ-finite measures \mu and \nu on a measurable space (X, \Sigma). Then \mu is said to be discrete with respect to \nu if there exists an at most countable subset S \subset X in \Sigma such that # All singletons \ with s \in S are measurable (which implies that any subset of S is measurable) # \nu(S)=0\, # \mu(X\setminus S)=0.\, A measure \mu on (X, \Sigma) is discrete (with respect to \nu) if and only if \mu has the form :\mu = \sum_^ a_i \delta_ with a_i \in \mathbb_ and Dirac measures \delta_ on the set S=\_ defined as :\delta_(X) = \begin 1 & \mbox s_i \in X\\ 0 & \mbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
