From: "r.e.s." Subject: Re: 3 Sided Coin Date: Sat, 26 Jun 1999 10:48:33 -0700 Newsgroups: sci.math LMWapner wrote ... : Is there a mathematical way to construct a : right circular cylinder to function : as a fair coin with P(h)=P(t)=P(s)=1/3? : : I know of know precise way of doing this : other than by experimental trials : (empirically). Of course this isn't what you had in mind, but virtually any 3-sided object can function that way. Label its sides H,T,S, and toss it repeatedly (independently) in triplets of tosses, until getting a triplet in one of the following sets: h={HTS, HST}, t={THS,TSH}, s={SHT,STH}. This guarantees that p(h)=p(t)=p(s), for *any* positive p(H),p(T),p(S), because each of the triplet possibilities has the same unknown probability p(H)*p(T)*p(S). You may need to toss for quite a while though, since the mean number of tosses until getting (h or s or t) is E[N]=1/[6*p(H)*p(T)*p(S)]. (E[N]>=27/6=4.5, attaining this minimum only when p(H)=p(T)=p(S), and can obviously be arbitrarily large unless there is a reasonable degree of symmetry.) I think it's worth noting how we can start with probabilities that are "unknown" due to a lack of symmetry of the *object*, and by *our* symmetric behavior end up with exactly known probabilities. -- r.e.s. (Spam-block=XX) ============================================================================== From: "William L. Bahn" Subject: Re: 3 Sided Coin Date: Sat, 26 Jun 1999 15:46:28 -0700 Newsgroups: sci.math It seems like if you take a piece of triangular bar stock and round the ends (symmetrically, of course) so that it is so unstable that (in practice) it can't be balanced then you have driven the odds of it landing (and staying) on end to zero - or at least as low (and probably significantly lower) than it landing and staying on any other edge which, in theory, is always possible. JZS wrote in message <7l2go9$jaq$1@news-02.meganews.com>... >I was wondering if there is a known mathematical way to construct a fair >3-sided coin? >That is a coin that is cyclinderical in shape, so that: > >P(Head)=P(Tail)=P(Side) = 1/3 > >So that: > >P(Head)+P(Tail)+P(Side) = 1 > >I wrote an essay on it, and from experimental results, it seems that the >only way to do this is to construct a coin so that the height of the >cylinder equals the radius of it. > >Any help is appreciated. > >-JZS > >-- >dfgdfg@sdergdfg.com > > ============================================================================== From: "JZS" Subject: Re: 3 Sided Coin (essay) Date: Sat, 26 Jun 1999 21:07:10 -0700 Newsgroups: sci.math here is the essay I wrote about it. Comments are welcome! __ Three Sided Coins We are all very familiar with two-sided coins. We carry them around everyday in our pockets, feed them to videogames, and some of us even use them (by flipping) to make important decisions. Additionally, we know that a fair two-sided coin has equal probability (50%) of landing on either its side designated Heads, or its other side, Tails. Would this same idea be possible with a three-sided coin? What would a three-sided coin look like? It would be similar to a two-sided coin, but would have a greater height, thus creating a third side (Edge). When trying to determine how to construct a fair three-sided coin, it is important to consider the ratio of height to diameter (H:D). It is very rare for a two-sided coin to land on its edge; conversely, if you were to flip a poster tube, it would almost always land on its edge, rather than either side. Therefore, one must determine the ratio of height to diameter that makes the probability of landing on any of the three sides of the coin uniform. If the ratio H:D is very small, the edge landing probability approaches 0, as in the example of the two-sided coin. If the ratio H:D is very large, then the probability of landing on the edge approaches 1, as in the case of the poster tube. I , and a friend set out to determine what H:D ratio would be required to create a fair three-sided coin. David made three coins in his machine shop. The coins were made smooth on all sides, to reduce uneven distributions of mass, which could slightly bias the outcomes. The ratios of H:D were as follows: .375, .5, and .667. Using our experimental data, we attempted to determine the edge landing probability as a function of the ratio H:D. Our statistical observations were used to establish the associated probabilities. Generally, statistical estimates for probabilities are valid if the experiment can be repeated a number of times under similar circumstances. In this case, repetition was possible. How does one flip a three-sided coin? A two-sided coin only has one main axis of rotation. However, a three-sided coin has two axes: the axis of the cylinder, and the axis orthogonal to that. Until the correct ratio H:D is determined (to create a fair coin), we certainly can’t flip the three-sided coin the same as the two-sided coin. Therefore, I flipped the three-sided coin on a line between the two axes, reasoning this would be a fair way to continue. A fair three-sided coin would have P(Head)+P(Tail)+P(Edge) = 1 -- the landing probabilities would be uniformly distributed. On July 30th, 1998, I flipped each of the three coins 3,000 times, which yielded percentages for each coin landing on Edge. When H:D = .375, Edge was obtained 4.6%. When H:D = .5, Edge was obtained 32.6%, and when H:D = .667, Edge was obtained 43.3%. The coin that fit the theoretical model the best was H:D = .5, or to write it differently when H=Radius. In general, symmetry often suggests a uniform distribution function, as was the case with this experiment. Coins have a wide variety of uses, and three-sided coins could have ramifications in several areas. Coins are often used to simulate random number generators, and a three-sided coin could add something extra to sports and gaming. Sports that begin with the flip of a coin to determine possession (i.e. football) would be impacted because the opportunity exists for the teams to not select the side that comes up (two teams, three sides). Does the referee then determine who gets the ball? Is the decision based on fan input? With a three-sided coin, perhaps Edge would mean that someone other than the two teams makes the decision. A three-sided gaming coin would certainly alter the winning percentages in Las Vegas! And three-sided coins would very dramatically impact the designs of coin operated machinery. I am pleased with the results of this experiment, and plan to repeat it in the future with some modifications. Variables to consider might include rate of spin, density and composition of coin, flipping surface, and coin design (i.e., smooth versus serrations). While three-sided coins may never be used on a daily basis, they create some intriguing possibilities with statistical probabilities. ============================================================================== From: "William L. Bahn" Subject: Re: 3 Sided Coin Date: Sun, 27 Jun 1999 17:29:40 -0700 Newsgroups: sci.math Could you please provide support for this claim? It seems like an intuitively reasonable claim, but it falls apart when you consider the behavior of real coins. For this claim to be true, real coins have to have an appreciable probability of landing and staying on their edges. The same is true for rods. For instance, a nickel is approximately 1/16" think and 13/16" in diameter. According to this claim, it should have better than a 7.6% chance of coming up edges. If it was even a fraction of one percent, it would happen frequently enough that whenever someone said "Heads or tails?" people would respond, "What if it comes up edges?" You can bet (no pun intended) that Las Vegas would take advantage of this possibility (just like the green spot in roulette). If real behavior departs so drastically from theory, the theory is either wrong or incomplete. If it is this incomplete, then it must be enhanced to make it useable. If you wish to claim that it is sufficiently accurate except for extreme radius to thickness to ratios (and a nickel only has a ratio of 6.5) then you really need to be able to describe the factors which become so dominant for those ratios that deviate from theory AND show that they are either driven to negligible levels or that they cancel out for the ratios you are working with. John Bailey wrote in message <37760680.1212649749@news.frontiernet.net>... >On Sat, 26 Jun 1999 16:20:09 -0700, "JZS" wrote: > >>>>I was wondering if there is a known >>>>mathematical way to construct a fair >>>>3-sided coin? That is a coin that is >>>>cyclinderical in shape, so that: >>>>P(Head)=P(Tail)=P(Side) = 1/3 >>>>So that: P(Head)+P(Tail)+P(Side) = 1 >> >>I was wondering what would be the best way with still preserving the *coin* >>shape. > >In that case, the coin's proportions must be such that an enscribing >sphere is cut into three equal areas by the intersection with the >edges of the coin. > >John ============================================================================== From: "Martin" Subject: Re: 3 Sided Coin (and von Neumann & Leech) Date: Wed, 30 Jun 1999 22:14:35 +0100 Newsgroups: sci.math You can find the problem and a brief discussion on 'Fifty Challenging Problems in Probability' by Frederick Mosteller but there is no mention of von Neumann and Leech. Nitsu wrote in message <7l5l3h$p04$1@news-02.meganews.com>... > >>This is a problem solved in several seconds by both >>von Neumann & by John Leech. >>Embed the coin in its circumscribing sphere with edges >>coincident with the "tropics". >>Project onto circumscribing vertical cylinder. >>Probabilities are proportional to areas. Here 1/3,1/3,1/3. >>If the radius of the sphere is 1, then the thickness >>of the coin is 2/3 and its diameter is 4sqrt(2)/3, >>thus giving a diameter: thickness ratio of 2sqrt(2). > > >Interesting! > >Where can I find this published? ============================================================================== From: "jzs" Subject: Re: PUZZLE: How to make fair die out of a convex polyhedron Date: Sat, 25 Sep 1999 08:16:12 -0700 Newsgroups: rec.games.board,sci.stat.math,sci.math >> T = D / tan(60deg) = D / sqrt(3) ~= 0.577 * D >>It couldn't be that simple, could it? >No - because surely it will depend a lot on how it is thrown and where >the spin axis is. >I think this would be solved by experiment before it was solved by >calculation... See: http://revolution.martini.nu/3Sided.doc ..for my experimental results. However, I am not saying that that is the correct result...just how my experiment came out! :) (basically that the height of the coin = the radius of the coin) ============================================================================== From: "Moo K. Chung" Subject: Re: PUZZLE: How to make fair die out of a convex polyhedron Date: Sun, 26 Sep 1999 00:40:36 GMT Newsgroups: rec.games.board,sci.stat.math,sci.math Obviously we need a certain condition to meaningfully solve it without resorting to physical experiments. The small booklet by F. Mosteller(You can buy it from Dover for C$7.50) solved the problem under the following condition: "... the coin might be tossed in such a way that it fell on a thick sticky substance that would grip the coin when it touched, and then the coin would slowly settle to its face." Oh by the way, the book tells me very interesting story about this problem. "On first hearing this question, John von Neumann, was unkind enough to solve it - including a 3-decimal answer - in his head in 20 seconds in the presence of some unfortunates who labored much longer." :) There is another famous puzzle relating Neumann's brute computational power involving a bee traveling back and forth between two colliding trains. The question was to determine the distance the bee has to travel until the two trains collides. Upon hearing the problem, after only a few second, he solved the problem. So the questioner asked him if he solved by that intuitive way of thinking the total flight time instead of the total distance the bee has to fly. Neumann replyed that he solved it by computing the sum of the infinite series of distances! What a man! ============================================================================== [For some experimental data see also 93_back/3side