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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 ...
Risk of ruin is a concept in gambling, insurance, and finance relating to the likelihood of losing all one's investment capital or extinguishing one's bankroll below the minimum for further play. [1] For instance, if someone bets all their money on a simple coin toss, the risk of ruin is 50%.
If the Markov chain is time-homogeneous, then the transition matrix P is the same after each step, so the k-step transition probability can be computed as the k-th power of the transition matrix, P k. If the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution π. [41]
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 ...
A game of snakes and ladders or any other game whose moves are determined entirely by dice is a Markov chain, indeed, an absorbing Markov chain. This is in contrast to card games such as blackjack, where the cards represent a 'memory' of the past moves. To see the difference, consider the probability for a certain event in the game.
In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a weak law of large numbers without the independence assumption, [296] [297] [298] which had been commonly regarded as a requirement for such ...
Gambler's fallacy; Inverse gambler's fallacy; Parrondo's paradox; Pascal's wager; Gambler's ruin; Poker probability. Poker probability (Omaha) Poker probability (Texas hold 'em) Pot odds; Roulette. Martingale (betting system) The man who broke the bank at Monte Carlo; Lottery. Lottery machine; Pachinko; Coherence (philosophical gambling ...
Note that even food, clothing, and shelter can be considered fixed transaction costs and thus contribute to the gambler's probability of ultimate ruin. This equation was the first application of Shannon's theory of information outside its prevailing paradigm of data communications (Pierce).