QuoteOriginally posted by: Traden4AlphaIndeed! Given that the financial markets are "ultra-realistic" with the trillions of dollars/euros/drachmas and the lives of millions/billions at stake, that seems warranted.What's crucial here is the breaking of an unstated central assumption. In the academic world, all coins are fair unless stated or until proven otherwise. In the real world, no coins are fair until proven otherwise. And given the limits of finite sample sizes, one can never prove that a real-world coin is fair, only that it is unlikely to be unfair outside a confidence interval.I totally agree Traden, but the sad thing is that we have to treat some problems in a irrealistic way (IMHO ).Now you questioned the existence of a fair coin...I recently questioned the fairness of mathematical models in Finance...Are these stuffs really related to Brownian motion or more advanced levy process? (Yes, they look like so , that's how my teachers convince me, but how to say "yes" with 100% confidence since we only have 1 sample path without access to a "parallel world"?)And the calibration calibrate to what...the real market process or the imagined process in most traders/dealers' head (by adding primiums according to their own likes/dislikes?) ?To be honest I am a noob but when I thought about this...I am having a head-ache...and feel super sad. Then I hold my little white flag (to surrender) and admit that mathematics only explains (or at least trying to) a part of the facts with perfect models which will simplify our analysis...(here comes the Brownians, martingales, optimal stopping times, risk-neutral pricing...etc etc). After all...what's a fair price in the real world?Just some complaints after one bad day of stucking in my project... EDIT: and I think a fair coin is a simple model for teaching purpose, though I do believe in it without thinking when randomizing my choices in Multiple-Choice Exams...

QuoteOriginally posted by: foxkingdomQuoteOriginally posted by: Traden4AlphaIndeed! Given that the financial markets are "ultra-realistic" with the trillions of dollars/euros/drachmas and the lives of millions/billions at stake, that seems warranted.What's crucial here is the breaking of an unstated central assumption. In the academic world, all coins are fair unless stated or until proven otherwise. In the real world, no coins are fair until proven otherwise. And given the limits of finite sample sizes, one can never prove that a real-world coin is fair, only that it is unlikely to be unfair outside a confidence interval.I totally agree Traden, but the sad thing is that we have to treat some problems in a irrealistic way (IMHO ).Now you questioned the existence of a fair coin...I recently questioned the fairness of mathematical models in Finance...Are these stuffs really related to Brownian motion or more advanced levy process? (Yes, they look like so , that's how my teachers convince me, but how to say "yes" with 100% confidence since we only have 1 sample path without access to a "parallel world"?)And the calibration calibrate to what...the real market process or the imagined process in most traders/dealers' head (by adding primiums according to their own likes/dislikes?) ?To be honest I am a noob but when I thought about this...I am having a head-ache...and feel super sad. Then I hold my little white flag (to surrender) and admit that mathematics only explains (or at least trying to) a part of the facts with perfect models which will simplify our analysis...(here comes the Brownians, martingales, optimal stopping times, risk-neutral pricing...etc etc). After all...what's a fair price in the real world?Just some complaints after one bad day of stucking in my project... EDIT: and I think a fair coin is a simple model for teaching purpose, though I do believe in it without thinking when randomizing my choices in Multiple-Choice Exams...Very true! Coins don't make a profit from appearing to be fair or unfair.Now imagine a coin that we flip and if we see too many heads in a row, we take actions that change the coin. Either we pile on to the heads side and make heads much more likely. Or we short the heads side on the expectation of a reversion. Is the coin fair?

QuoteOriginally posted by: Traden4Alpha Is the coin fair?I think here we might need a new definition of fairness. (or I am too tired to conduct reasonable thinking...) What do you think?

QuoteOriginally posted by: foxkingdomQuoteOriginally posted by: Traden4Alpha Is the coin fair?I think here we might need a new definition of fairness. (or I am too tired to conduct reasonable thinking...) What do you think?Although we usually think of a fair coin as being exactly 50:50, the real prerequisite for fairness seems to be the knowability of the probability of some future event. If one knew that a particular coin would show "heads" with p=0.5200000, then one could adjust the pay-off structure in a fair way.In turn, a knowable probability requires: 1) enough empirical data to estimate the probability; 2) justification that ∂p(t)/∂t = 0 (or that the time-rate-of-change of probability has a knowable value). The first requirement implies that we need sufficient experience with the system. The second requirement implies that no one can influence the probability (or pay-offs) for self-serving purposes. In the context of financial markets, the first requirement means having enough historical data over a sufficiently wide-range of economic conditions and the second requirement means that market manipulation is unlikely and that all parties have the financial resources to meet their obligations.

QuoteOriginally posted by: outrunThat's a mean reverting coin with negative expected dP(H)/(P(H)-1/2) !Or its a speculative bubble coin with self-amplifying tendencies toward 100% H or 100% T (was Haldane right?????).

If I flip an euro coin continuously for 10 years, and I get an euro coin every time, what is the probability of getting a slightly different euro coin and a drachma banknote on the next flip ?

I think we have to start with a prior probability of coing being fair, not with prior probability for heads/tails!Having said that, the problem cannot be simplified to just "having 100 tosses". Need to define what exactly we know before observing the outcomes and how do we quantify our knowledge. Second point is that the coin may be fair but tossed in a certain way - fairness of coin is not a quality per se.For example, have we looked at the coin beforehand and noticed that both tail and head are present (i.e. not both heads)?So, one needs to choose a framework or a prism through which she looks at it. Need to have some particular physical connections, probably.Or turn this task into learning of what the interviewer actually means by fair coin. probe with questions: does it matter if we are talking about coins or cards or balls or markets. Say, is your task equivalent to "what is the probability that the urn contains only black balls all the way?"

QuoteOriginally posted by: outrunGreat!So far we have:* It's impossible to be certain on fairness based on a finite number of samples.* The coin will be affected by flipping it. It deforms, scratches, etc.* It's impossible to construct a fair coin prior to tossing it. The Heisenberg uncertainty principle prevents you from inspecting the coin for possible needed alterations without distorting it.* The fairness is not only a function of the coin, but also the toss. I bet some people at MIT can make a fair coin unfair with trajectory prediction & control* An Euro coin may not always stay the same Euro coin.What about computability and representability?Suppose we were able to do an infinite number of tosses, we still wouldn't be able to represent the accumulated p value with a finite amount of physical memory.Worse, the cumulative statistics don't prove fairness at all.The coin's output might be governed by a deterministic process such as IF(X_i>c, "Heads", "Tails") where X_i+1 = 4*(X_i)*(1-X_i) or some other simple PRNG. The result might look fair, but be entirely unfair in the sense that a clever observer might deduce the pattern and become able to perfectly predict the next coin state.Perhaps a fair coin is like pi: it exists as a logical, but not a physical construct. It's a useful approximation, but one should never bet too much on it being true to the last decimal point or even the 5th decimal point.