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  2. Gambler's ruin - Wikipedia

    en.wikipedia.org/wiki/Gambler's_ruin

    In statistics, gambler's ruin is the fact that a gambler playing a game with negative expected value will eventually go bankrupt, regardless of their betting system.. The concept was initially stated: A persistent gambler who raises his bet to a fixed fraction of the gambler's bankroll after a win, but does not reduce it after a loss, will eventually and inevitably go broke, even if each bet ...

  3. Risk of ruin - Wikipedia

    en.wikipedia.org/wiki/Risk_of_ruin

    for a random walk with a starting value of s, and at every iterative step, is moved by a normal distribution having mean μ and standard deviation σ and failure occurs if it reaches 0 or a negative value. For example, with a starting value of 10, at each iteration, a Gaussian random variable having mean 0.1 and standard deviation 1 is added to ...

  4. Markov chain - Wikipedia

    en.wikipedia.org/wiki/Markov_chain

    In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event.

  5. Optional stopping theorem - Wikipedia

    en.wikipedia.org/wiki/Optional_stopping_theorem

    Then the gambler's fortune over time is a martingale, and the time τ at which he decides to quit (or goes broke and is forced to quit) is a stopping time. So the theorem says that E[X τ] = E[X 0]. In other words, the gambler leaves with the same amount of money on average as when he started. (The same result holds if the gambler, instead of ...

  6. Matrix analytic method - Wikipedia

    en.wikipedia.org/wiki/Matrix_analytic_method

    [1] [2] Such models are often described as M/G/1 type Markov chains because they can describe transitions in an M/G/1 queue. [3] [4] The method is a more complicated version of the matrix geometric method and is the classical solution method for M/G/1 chains. [5]

  7. Gambling and information theory - Wikipedia

    en.wikipedia.org/wiki/Gambling_and_information...

    When these constraints apply (as they invariably do in real life), another important gambling concept comes into play: in a game with negative expected value, the gambler (or unscrupulous investor) must face a certain probability of ultimate ruin, which is known as the gambler's ruin scenario. Note that even food, clothing, and shelter can be ...

  8. Stochastic matrix - Wikipedia

    en.wikipedia.org/wiki/Stochastic_matrix

    The stochastic matrix was developed alongside the Markov chain by Andrey Markov, a Russian mathematician and professor at St. Petersburg University who first published on the topic in 1906. [3] His initial intended uses were for linguistic analysis and other mathematical subjects like card shuffling , but both Markov chains and matrices rapidly ...

  9. Discrete-time Markov chain - Wikipedia

    en.wikipedia.org/wiki/Discrete-time_Markov_chain

    A Markov chain with two states, A and E. In probability, a discrete-time Markov chain (DTMC) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past.