Wikipedia:Reference desk/Archives/Mathematics/2020 February 4
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February 4[edit]
Likeliest number of chess sets[edit]
The Lewis chessmen hoard contains 60 major chess pieces: 8 kings, 8 queens, 16 bishops, 15 knights and 13 rooks (which are called warders). They are listed here. They show no signs of colour. There are also 19 pawns so ignore them. If they came from four chess sets (as is often supposed) there would be 1 knight and 3 rooks missing but the individual pieces vary a lot from each other and so any definite allocation to sets is not possible. Some experts think that, on stylistic grounds, they may have come from five sets (or even more, but a small number anyway). It seems to me intuitively that it would be very unlikely to get such a distribution of pieces if they had come at random from five or more sets. Has this been looked at statistically? Thincat (talk) 09:53, 4 February 2020 (UTC)
- I'm not aware of any such study and it would surprise me if there was one. I don't see any reason the pieces would have been taken at random (i.e. as independent draws in an urn model with urns consisting of chess sets). They'd more likely be correlated. 2601:648:8202:96B0:0:0:0:E118 (talk) 10:24, 4 February 2020 (UTC)
- I think that for statistical modelling purposes it is a reasonable assumption that the pieces that went missing were like random draws from an urn with pieces, originally consisting of n complete sets of 16 officers. --Lambiam 14:13, 4 February 2020 (UTC)
- Under that assumption, I compute the probability that four complete sets of officers, after losing 4 pieces, end up with the composition of the hoard as being equal to exactly 0.01410188612726952. For five complete sets of officers from which 20 pieces were lost, I find approximately 0.0033354 (or, exactly, 147396219858000/44191451777652179). The latter is roughly one quarter as likely, not dramatic enough a reduction to override considerations of style. For six sets I get about 0.0018358. It is furthermore conceivable that at some time complete sets were used for playing, which however were already heterogeneous because missing pieces had been replaced by pillaging artistically less accomplished and possibly also incomplete seta. --Lambiam 16:25, 4 February 2020 (UTC)
- Thank you for those calculations. As you say, it doesn't make five sets look very much less plausible than four. Pondering on your reply, I think I now have a clearer view of what I should have asked this morning. Consider five complete sets of major pieces – 80 pieces – where all the pieces of each type are identical (and the same colour). We draw out 60 pieces. What is the chance we have 8 or less kings and queens and 16 or less of each of the other types of piece? Now, very much an aside – For nearly 200 years only 59 pieces were known until last year another piece, probably from the hoard, turned up. If it had been a king, queen or bishop the four set theory would have been disproved. (Or, more likely, people would have said the new piece wasn't from the hoard at all.) In fact it was a warder (rook) so everyone was happy, especially the person who was then able to sell it for £735,000! Thincat (talk) 19:43, 4 February 2020 (UTC)
- The new condition includes all previously included compositions and many, many more, so it should not be surprising that it comes out much higher. I get about 0.22020. --Lambiam 20:36, 4 February 2020 (UTC)
- Thank you again. Going back to my elementary statistics, I think that suggests that, given a null hypothesis of five sets, I should not be rejecting that on grounds of probability. I accept, of course, that the loss of pieces may have been selective and not random. Thincat (talk) 09:11, 5 February 2020 (UTC)
- The new condition includes all previously included compositions and many, many more, so it should not be surprising that it comes out much higher. I get about 0.22020. --Lambiam 20:36, 4 February 2020 (UTC)
- Thank you for those calculations. As you say, it doesn't make five sets look very much less plausible than four. Pondering on your reply, I think I now have a clearer view of what I should have asked this morning. Consider five complete sets of major pieces – 80 pieces – where all the pieces of each type are identical (and the same colour). We draw out 60 pieces. What is the chance we have 8 or less kings and queens and 16 or less of each of the other types of piece? Now, very much an aside – For nearly 200 years only 59 pieces were known until last year another piece, probably from the hoard, turned up. If it had been a king, queen or bishop the four set theory would have been disproved. (Or, more likely, people would have said the new piece wasn't from the hoard at all.) In fact it was a warder (rook) so everyone was happy, especially the person who was then able to sell it for £735,000! Thincat (talk) 19:43, 4 February 2020 (UTC)
- Under that assumption, I compute the probability that four complete sets of officers, after losing 4 pieces, end up with the composition of the hoard as being equal to exactly 0.01410188612726952. For five complete sets of officers from which 20 pieces were lost, I find approximately 0.0033354 (or, exactly, 147396219858000/44191451777652179). The latter is roughly one quarter as likely, not dramatic enough a reduction to override considerations of style. For six sets I get about 0.0018358. It is furthermore conceivable that at some time complete sets were used for playing, which however were already heterogeneous because missing pieces had been replaced by pillaging artistically less accomplished and possibly also incomplete seta. --Lambiam 16:25, 4 February 2020 (UTC)
- The situation is similar to the German tank problem (not sure why it was not linked yet): doing the math is the easy part, but choosing the initial hypotheses is hard/controversial because Bayes' theorem requires a prior probability.
- For instance, reading the article's account of the discovery of the pieces, I would wildly speculate that the sets were complete or almost complete (trader's stock, never used) and buried in the sand dune, the 1831 guy dug up all he could but missed a few, someone else re-dug the same dune and found the rest, of which one recently resurfaced. If that story is true (that's a big if) it would make sense that only few pieces were missed in the first dig, so that weighs in favor of the four-set theory (missing 4 pieces) instead of the 5-set theory (missing 20 pieces). In the Bayesian framework, that would mean lending more initial credence to the four-set theory than to the five-set theory before doing the calculations above (which amplify the effect).
- My narrative is a bit silly but the point is you have to consider what narrative your statistics are based on. TigraanClick here to contact me 16:51, 5 February 2020 (UTC)