Abstract
Some uncertainty about flipping a biased coin can be resolved from the sequence of coin sides shown already. We report the exact amounts of predictable and unpredictable information in flipping a biased coin. Fractional coin flipping does not reflect any physical process, being defined as a binomial power series of the transition matrix for “integer” flipping. Due to strong coupling between the tossing outcomes at different times, the side repeating probabilities assumed to be independent for “integer” flipping get entangled with one another for fractional flipping. The predictable and unpredictable information components vary smoothly with the fractional order parameter. The destructive interference between two incompatible hypotheses about the flipping outcome culminates in a fair coin, which stays fair also for fractional flipping.
Highlights
The vanishing probability of winning in a long enough sequence of coin flips features in the opening scene of Tom Stoppard’s play “Rosencrantz and Guildenstern Are Dead”, where the protagonists are betting on coin flips
Coin-tossing experiments are ubiquitous in courses on elementary probability theory, and coin tossing is regarded as a prototypical random phenomenon of unpredictable outcome, the exact amounts of predictable and unpredictable information related to flipping a biased coin was not discussed in the literature
We show that the side repeating probabilities considered independent of each other in the standard, “integer” coin-tossing model appear to be entangled with one another as a result of strong coupling between the future states in fractional flipping
Summary
The vanishing probability of winning in a long enough sequence of coin flips features in the opening scene of Tom Stoppard’s play “Rosencrantz and Guildenstern Are Dead”, where the protagonists are betting on coin flips. We propose the information theoretic study of the most general models for “integer” and fractional flipping a biased coin. We show that these stochastic models are singular (along with many other well-known stochastic models), and their parameters—the side repeating probabilities—cannot be inferred from assessing frequencies of shown sides (see Sections 2 and 4). We study the evolution of the predictable and unpredictable information components of entropy in the model of fractional flipping a biased coin (Section 5).
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