Related papers: A Coin-Tossing Conundrum
What is the relationship between plausibility logic and the principle of maximum entropy? When does the principle give unreasonable or wrong results? When is it appropriate to use the rule `expectation = average'? Can plausibility logic…
Consider the following game: You are given two indistinguishable envelopes, each containing money. One contains twice as much as the other. You may pick one envelope and keep the money it contains. Having chosen an envelope, you are given…
Consider the following probability puzzle: A fair coin is flipped n times. For each HT in the resulting sequence, Bob gets a point, and for each HH Alice gets a point. Who is more likely to win? We provide a proof that Bob wins more often…
Have you ever taken a disputed decision by tossing a coin and checking its landing side? This ancestral "heads or tails" practice is still widely used when facing undecided alternatives since it relies on the intuitive fairness of…
Probability forecasts are intended to account for the uncertainties inherent in forecasting. It is suggested that from an end-user's point of view probability is not necessarily sufficient to reflect uncertainties that are not simply the…
The standard textbook method for estimating the probability of a biased coin from finite tosses implicitly assumes the sample sizes are large and gives incorrect results for small samples. We describe the exact solution, which is correct…
Finding a counterfeit coin with the different weight from a set of visually identical coin using a balance, usually a two-armed balance, known as the balance question, is an intersting and inspiring question. Its variants involve…
Parrondo's coin-tossing games comprise two games, $A$ and $B$. The result of game $A$ is determined by the toss of a fair coin. The result of game $B$ is determined by the toss of a $p_0$-coin if capital is a multiple of $r$, and by the…
In this article, I will present a paradox whose purpose is to draw your attention to an important topic in finance, concerning the non-independence of the financial returns (non-ergodic hypothesis). In this paradox, we have two people…
We review the quantum version of a well known problem of cryptography called coin tossing (``flipping a coin via telephone''). It can be regarded as a game where two remote players (who distrust each other) tries to generate a uniformly…
Exchangeability is a central notion in statistics and probability theory. The assumption that an infinite sequence of data points is exchangeable is at the core of Bayesian statistics. However, finite exchangeability as a statistical…
This paper examines the biases and performance of several uncertain inference systems: Mycin, a variant of Mycin. and a simplified version of probability using conditional independence assumptions. We present axiomatic arguments for using…
Getting an unbiased result is a remarkably long standing problem of collective observation/measurement. It is pointed out that quantum coin tossing can generate unbiased result defeating dishonesty.
The hypothesis of randomness is fundamental in statistical machine learning and in many areas of nonparametric statistics; it says that the observations are assumed to be independent and coming from the same unknown probability…
We consider the urn setting with two different objects, ``good'' and ``bad'', and analyze the number of draws without replacement until a good object is picked. Although the expected number of draws for this setting is a standard textbook…
I defend an analog of probabilism that characterizes rationally coherent estimates for chances. Specifically, I demonstrate the following accuracy-dominance result for stochastic theories in the C*-algebraic framework: supposing an…
Inspired by the theory of desirable gambles that is used to model uncertainty in the field of imprecise probabilities, I present a theory of desirable things. Its aim is to model a subject's beliefs about which things are desirable. What…
The problem of creating a three-sided dice with the probability of it landing on each of its sides being equal to 1/3 has been around for many years. Various approaches have been attempted, but as different authors achieved at different…
It is well known that there is no direct one-to-one relation between $p$-values and likelihood ratios or Bayes factors, since their relation crucially involves the sample size $n$. We investigate their (asymptotic) relation in a…
Various measures can be used to estimate bias or unfairness in a predictor. Previous work has already established that some of these measures are incompatible with each other. Here we show that, when groups differ in prevalence of the…