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A gambler's fortune (capital) is a martingale if all the betting games which the gambler plays are fair. The gambler is playing a game of coin flipping. Suppose X n is the gambler's fortune after n tosses of a fair coin, such that the gambler wins $1 if the coin toss outcome is heads and loses $1 if the coin toss outcome is tails. The gambler's ...
Consider a simple statistical model of a coin flip: a single parameter that expresses the "fairness" of the coin. The parameter is the probability that a coin lands heads up ("H") when tossed. can take on any value within the range 0.0 to 1.0. For a perfectly fair coin, =. Imagine flipping a fair coin twice, and observing two heads in two ...
Tossing a coin. Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking which side is showing when it lands, in order to randomly choose between two alternatives. It is a form of sortition which inherently has two possible outcomes. The party who calls the side that is facing up when the coin ...
The entropy of the unknown result of the next toss of the coin is maximized if the coin is fair (that is, if heads and tails both have equal probability 1/2). This is the situation of maximum uncertainty as it is most difficult to predict the outcome of the next toss; the result of each toss of the coin delivers one full bit of information.
In probability theory, stochastic drift is the change of the average value of a stochastic (random) process. A related concept is the drift rate, which is the rate at which the average changes. For example, a process that counts the number of heads in a series of fair coin tosses has a drift rate of 1/2 per toss. This is in contrast to the ...
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 coins would have /
In this example, the probability of losing the entire bankroll and being unable to continue the martingale is equal to the probability of 6 consecutive losses: (10/19) 6 = 2.1256%. The probability of winning is equal to 1 minus the probability of losing 6 times: 1 − (10/19) 6 = 97.8744%. The expected amount won is (1 × 0.978744) = 0.978744.
The St. Petersburg paradox or St. Petersburg lottery [1] is a paradox involving the game of flipping a coin where the expected payoff of the lottery game is infinite but nevertheless seems to be worth only a very small amount to the participants. The St. Petersburg paradox is a situation where a naïve decision criterion that takes only the ...