March 2016 - mathfun@mailman.xmission.com

[math-fun] how to test whether a coin is fair?
by Warren D Smith 04 Mar '16

04 Mar '16

> WDS: > You have a coin. You want to know whether it is fair. > An unbounded testing procedure is as follows: > > for(N=1,2,3,...){ > Toss coin. Update head and tails counts H & T (with H+T=N). > If( |H-T| > F(N,K) ){ output "coin is unfair with confidence>1-K" and > stop. } > } > > The key is the magic function F(N,K). > What should it be? > > Roughly speaking I think it ought to be > F(N,K) =approx= sqrt(2*N*(C*ln(1/K)+ln(A+ln(B+N)))) > where A and B and C are appropriate positive constants. > Using A=7, B=2, and C=1 seems to work not too horribly > provided N>=8. But I do not know what the best values of A,B,C are > nor whether this F(N,K) formula really has much validity. --the following seems to work better: A=8, B=3, C=1. > From: Eugene Salamin <gene_salamin(a)yahoo.com> > This can be cast as a Bayesian probability. Let x be the probability of > heads. We need a prior probability for x, which I will just assume is > uniform. Then the posterior probability for x is > > P(x) = C x^h (1-x)^t, > > where C is a normalizing constant. The matter of whether the coin is "fair" > lies outside of probability. It is personal opinion, or, being fancy and > minimizing risk over the consequences, is decision theory. --Well, not quite. First, the whole "personal opinion" jive is only if we are willing to throw all of the theory of designing statistical tests, in the garbage as all bullshit. I'm not. I like your idea of trying to be Bayesian with uniform prior. I personally had been being nonBayesian and thinking about "confidence bounds." However, my test was NOT for one specific number of coin tosses and hence one specific (H,T). It was for keeping on tossing coins forever until one day (maybe) your unfairness criterion is exceeded, whereupon you stop and report the coin is unfair with some "confidence" >=1-K. If it is a fair coin, then with probability>=1-K that day will never come. So basically, the definition of K can be back-deduced from the definition of F(N,K) by asking "what is the probability this procedure will terminate?" Answer P should equal 1-K. If it does not, then our definition of F(N,K) was not quite correct. Mine undoubtably is not quite correct, but seems to come close in the sense that P appears to approximately equal 1-K throughout the range 0.0001 <= K <= 0.25. Specifically with A=8, B=3, and all N from 4 to 10000, the correct value of C always appears to be in [0.91, 1.05]. I'm not sure if what I just said about P=1-K uniquely determines F(N,K). If there still is freedom of choice, then we want the prescription yielding the "most power"... meaning that if the coin is biased, then the expected wait time before detection, is minimized. This if we assume a prior (such as Eugene Salamin's uniform prior, or one could use others) on the coin bias x, is a well defined quantity for any particular correct F(N,K). > From: Kerry Mitchell <lkmitch(a)gmail.com> > It seems to me that |H-T| will tend to infinity as N goes to infinity > (albeit at a slower rate), --true. In fact the "law of the iterated logarithm" says that for a fair coin, with probability=1, we have lim sup |H-T| / sqrt(2*N*lnlnN) = 1 when N=total # coin tosses --> infinity, and H and T are functions of N=1,2,3,... I.e. this limsup both exists with probability=1, and its value is exactly 1 with probability=1. My F(N,K) of course is based on that law, but goes further, offering a considerably more precise statement. Or at least, if would, if my F(N,K) were correct, or could be proven correct to within some known upper & lower bounds. > There's also the > question of the distribution of the flips--one could have a coin with the > perfect numbers of flips, but, over the long run, if the distribution were > exactly predictable (say, H, T, H, T, H, T, etc.), that would be a good > argument for the coin not being fair. --I will allow you to assume, for purposes of this problem, that all flips truly are independent. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] how to test whether a coin is fair?
by Warren D Smith 04 Mar '16

04 Mar '16

You have a coin. You want to know whether it is fair. An unbounded testing procedure is as follows: for(N=1,2,3,...){ Toss coin. Update head and tails counts H & T (with H+T=N). If( |H-T| > F(N,K) ){ output "coin is unfair with confidence>1-K" and stop. } } The key is the magic function F(N,K). What should it be? Roughly speaking I think it ought to be F(N,K) =approx= sqrt(2*N*(C*ln(1/K)+ln(A+ln(B+N)))) where A and B and C are appropriate positive constants. Using A=7, B=2, and C=1 seems to work not too horribly provided N>=8. But I do not know what the best values of A,B,C are nor whether this F(N,K) formula really has much validity. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Loose packing problem
by James Propp 04 Mar '16

04 Mar '16

How many billiard balls can be packed into a standard billiard rack if no two balls are allowed to touch? (Of course ten will fit if they are allowed to touch.) I can fit six, but I don't see how to prove that seven is impossible. (I can prove that in a smaller rack that could accommodate six touching balls, you can't fit more than four non-touching balls; it's a nice application of the pigeonhole principle.) Is there literature on this flavor of packing problem? Note that a question about packing non-touching disks of radius r can be paraphrased as a question of the form "Is it possible, for all epsilon > 0, to pack disks of radius r-plus-epsilon ..." (where the non-touching condition is dropped). Jim Propp

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[math-fun] non-reproducibility of science. specifically, psychology studies.
by Warren D Smith 03 Mar '16

03 Mar '16

A 2015 report by the "Open Science Collaboration," called the "Reproducibility Project," found that less than 40 out of a sample of 100 psychology papers in leading journals held up when retested by an independent team. http://science.sciencemag.org/content/351/6277/1037.2.full fights back, claiming that report "contains three statistical errors and provides no support for such a conclusion. Indeed, the data are consistent with the opposite conclusion, namely, that the reproducibility of psychological science is quite high." http://science.sciencemag.org/content/351/6277/1037.3.full returns fire, claiming that criticism's "very optimistic assessment is limited by statistical misconceptions and by causal inferences from selectively interpreted, correlational data. Using the Reproducibility Project: Psychology data, both optimistic and pessimistic conclusions about reproducibility are possible, and neither are yet warranted." Are we having fun yet? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] This Crazy Contraption Makes Beautiful Music Using Marbles
by Victor S. Miller 03 Mar '16

03 Mar '16

http://m.huffpost.com/us/entry/martin-molin-marble-machine-music_us_56d74ba… Sent from my iPhone

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[math-fun] Demis Hassabis, 5-time pentamind champion
by Warren D Smith 02 Mar '16

02 Mar '16

The "pentamind" is a competition where the goal is to place high in tournaments of 5 different kinds of mental games (for example it might be go, chess, poker, backgammon, and othello) happening at the "mind sports olympiad." It's sort of like the "decathlon" in physical sports, except the decathlon has 10 known standard events, whereas in pentamind the players get to choose which games to contest (subject to certain constraints, e.g. not allowed to do two "chess-like" games). Demis Hassabis, recently in the news about "AlphaGo," won this event 5 times, more than anybody else. It looks like a goodly percentage of the winners have something to do with mathematics. Un(?)fortunately it looks in recent years like the strategy used to win is to choose some rather unusual (silly?) games hardly anybody ever heard of. For example the games of "entropy," "triolet," "boku," and "snatch," whatever they are, "losing checkers," etc. There are too many games. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] black hole merger accompanied by gamma ray burst ??!!
by Keith F. Lynch 01 Mar '16

01 Mar '16

Allan Wechsler <acwacw(a)gmail.com> wrote: > As Warren points out, we want the gamma-ray source to be as close > to the disaster site as possible, and if it were *behind* the site, > we could get it to within 0.2 light-seconds. Nitpick: 0.18 light-seconds if you account for the cosmological red shift. > That increases the excitation by a factor of four (is that right?), Half the distance is four times the power but only twice the strain. > which isn't enough to overcome Warren's objections. I agree. Perhaps the event destabilized an orbit very close to the event, and the gammas were when that orbit decayed a few hundred milliseconds later. Perhaps a neutron star crashed into the new economy-size black hole, or was sent crashing into another neutron star or into a regular star. Or maybe it was a coincidence, and the two events weren't even in the same part of the sky. Or maybe the events were really simultaneous, but LIGO's clocks and the gamma ray observatory's clocks aren't synchronized. Perhaps one is on UTC and the other on UT0 or TAI. The most intriguing possibility is that gravitons have a small imaginary rest mass, hence always travel faster than c. If so, they could in principle be used to send messages back through time.

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[math-fun] Computational effort
by James Propp 01 Mar '16

01 Mar '16

I don't know what the relevant units of computation are --- floating-point operations are the ones I've heard about most, but what if you're not using floating point? --- but I'm wondering if anyone has tried to quantify "how much computing" took place in (a) code-breaking at Bletchley Park, (b) the Manhattan Project, and (c) the Apollo space program. It seems likely to me that my laptop, running some silly number-crunching problem in Mathematica and returning an answer after six hours that I could probably have figured out in my head if I'd thought a little bit more before jumping in to write some code, has them all beat. What do you think? (And while we're on the subject: What's the canonical example of wasted computational effort? Isn't there some well-known old story of someone who had an early computer calculate the determinant of a singular matrix?) Jim Propp

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