Re: Dice Problem
- Original Message - From: GEORGE PERKINS <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Thursday, June 29, 2000 2:07 PM Subject: Dice Problem > Recently a colleague came in the office with the following problem: > > Is there a way to 'load' two individual die so that all sums will be equally likely? > > (I take it that they would like to load the die in such a way that the sum of 2 is equally likely as a sum of 3, as a sum of 4...etc) > > Any ideas would be appreciated > GP If we specify that both dice shall be labelled 1,2,3,4,5,6, and that they shall roll independently, then the answer is "no". Proof: Let X,Y,S be the individual values and the sum. P(X=6)P(Y=6) = P(S=12) > 0 so P(X=6) and P(Y=6) are positive. Similarly P(X=1) and P(Y=1) are positive. P(X=1)P(Y=1) = P(S=2) = P(S=7) >= P(X=1)P(Y=6) + P(X=6)P(Y=1) > P(X=1)P(Y=6) Thus P(Y=1) > P(Y=6); but interchanging values we also prove the opposite inequality. Contradiction! If the dice are labelled in a nonstandard way, we can get (eg) 36 equally probable outcomes: (1,2,3,4,5,6) and (0,6,12,18,24,30) If the dice are in some way coordinated, so that the fall of one die influences the fall of the other, we can have any probabilities we like. -Robert Dawson === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: Dice Problem
>Recently a colleague came in the office with the following problem:>Is there a way to 'load' two individual die so that all sums will be equally likely? Although I doubt whether it is possible to load a pair of dice to produce results from a particular distribution it may be possible to load a single die. Obviously the uniform distribution required for George Perkins' original problem is easily achieved without having to load the dice. I have in front of me a die with twelve pentagonal faces that I bought from a Dungeons & Dragons gaming shop. (One can also get 4-sided, 8 sided, 20-sided and (most useful) ten sided dice). However, a non-uniform distribution of results (e.g., conforming to the binomial distribution) could be achieved by varying the area of the faces. The larger the area of a face the more likely the die will come to rest upon it. The centre of gravity of the die will no doubt be affected but I am sure somebody cleverer than me could figure out how to compensate for this. Graham *Dr Graham D. SmithPsychology DivisionPark CampusUniversity College NorthamptonBoughton Green Rd.NorthamptonNN2 7AL Tel: +44 (0) 1604 735500 Ext 2393E-mail: [EMAIL PROTECTED]*
Re: Dice Problem
On Thu, 29 Jun 2000, GEORGE PERKINS wrote: > Recently a colleague came in the office with the following problem: > > Is there a way to 'load' two individual die so that all sums will be > equally likely? > (I take it that they would like to load the die in such a way that the > sum of 2 is equally likely as a sum of 3, as a sum of 4...etc) If we are referring to standard cubical dice with 1, 2, ..., 6 pips on each face, the answer is "No." If the faces are renumbered, something like that would be possible, even without "loading" the dice. For a simple (some might say 'degenerate') example, consider a die with one pip on each face, and a second die with two pips on each face. The sum is invariably 3. For a more interesting example, with more than one possible sum, let one die have one pip on three faces, and two pips on the other three; and let the other die have two pips on three faces and four pips on the other three. The possible sums are 3, 4, 5, and 6, each of which can be obtained in any of 9 ways, and are therefore equally probable if the dice are "fair". No need to load them. More complex examples are clearly possible, and are left as exercises for the reader. -- Don. Donald F. Burrill [EMAIL PROTECTED] 348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED] MSC #29, Plymouth, NH 03264 603-535-2597 184 Nashua Road, Bedford, NH 03110 603-471-7128 === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Dice Problem
Recently a colleague came in the office with the following problem: Is there a way to 'load' two individual die so that all sums will be equally likely? (I take it that they would like to load the die in such a way that the sum of 2 is equally likely as a sum of 3, as a sum of 4...etc) Any ideas would be appreciated GP === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
