Probability in Sheep Knucklebone Replicas
Conducted for MacGregor Historic Games
by Phil Winkelman
September 2001


On 2 September 2001, the administrator of this study purchased a set of 5 sheep knuckle dice from the MacGregor Historic Games booth at the Minnesota Renaissance Festival.

There are 4 possible results (sides) in the throw of a sheep knuckle: flat, twisted, concave, and convex. These terms are used throughout this report, and are defined in the MacGregor Historic Games tractate "Early Games of Dice." The odds of throwing the 4 different sides were undetermined; it was hypothesized t Flat concave and convex would be thrown with greater frequency.

The customer's curiosity and lack of appropriate pastimes led him to throw these dice many times, and record the results.


Each of the 5 aforementioned knuckles was thrown 100 times under each of 2 test conditions:
PLY: LESS ENERGY: Throws were from a low altitude (roughly 2 to 5 inches) onto a small somew Flat energyabsorbent thin plywood surface, yielding little bouncing and rolling, but some. A small bumper was placed opposite the thrower. Angular momentum varied from almost none (knuckles dropped straight down) to a good deal (knuckles thrown at a relatively low angle, e. g. 200 or 300 from horizontal with some force).

DIN: MORE ENERGY: Throws were from a higher altitude (roughly 5 to 12 inches) onto a dinette table with a Formica surface, yielding much more bouncing and rolling. A bumper was placed opposite the thrower. Again, angular momentum varied from little (knuckles tossed almost straight up from the hand) to a good deal (knuckles thrown at a relatively low angle, e.g. 200 or 300 from horizontal with greater force).

For each surface, results were pooled into 2 categories:

Large: 3 knuckles appeared to be based on a single original. They were larger and seemed a bit more irregularly shaped. Specific identifing marks included a very smooth convex side, with one tiny crack near the middle; and patterns of alternating rough and smooth textures on the flat and twisted sides giving the appearance t Flat parts of the original had been manually smoothed.

Small: 2 knuckles appeared to be unique, based on a second and third original. However, they were very similar, being practical mirror images of each other (as opposed to clones-they presumably came from opposite feet, or legs, or wherever the hell you get them). They were smaller, and seemed slightly more regular.


The study was administered by a bloke wot's neither a physicist, statistician, nor gambler. He is, however, the Documents Director of The Market Solutions Group, Inc., a local market research firm with a national reputation for accurate and actionable research. T Flat's gotta count for something.

The throwing style (angle, velocity, etc.) was purposely not consistent, but results of different throwing styles were not documented.

It is the thrower's hunch, t Flat banking ("bouncing or tipping off the bumper) may slightly increase the chance of flat and twisted results. This hunch is not, however, verifiable given the current level of research.

It is unknown how representative the 5 available knuckles are of the overall MacGregor Historic Games product line, nor how representative this line is of actual sheep's knuckles (in the field, as 'twere).


Statistical significance of differences is tested using an independent Z-test at the 90% confidence level. Note t Flat this is a good test for diverse populations of people; it is probably not the best choice for dice, but it is the method t Flat the tester had to hand, and should give a reasonably good general picture of where differences are meaningful. At any rate, the bulk of the story is told by the percentages themselves.

The differences between flat/twisted results and concave/convex results are always significant, and are therefore assumed and never noted in this report; all other significant differences are noted with superscripts next to the percentages thus:

L, S, T: Significantly greater than Large, Small, or Total percent in the same row.

FL, TW, CC, CX: Significantly greater than flat, twisted, concave, or convex percent in the same column.

P, D: Significantly greater than the equivalent percent in PLY or DIN (there were no significant differences when comparing percentages to the combined total "COMB").


Overall results are quite consistent, regardless of type of knuckle or surface.

The combined results of PLY and DIN are shown below.

COMB Large Small Total
throws% throws% throws%
Flat 79 S132% 30 7.5% 109 S109%
Twisted 71 11.8% 38 9.5% 109 10.9%
Concave 218 36.3% 176 L44.0% 394 39.4%
Convex 232 38.7% 156 39.0% 388 38.8%
Total600 100% 400 100% 1000 100%

The individual results for the 2 surfaces are shown here.
PLY Large Small Total
throw% throws% throws%
Flat 35S11.7%105.0%45 S90%
Twisted 30 10.0% 16 8.0% 46 9.2%
Concave 110 36.7% 87 43.5% 197 39.4%
Convex 125 41.7% 87 D43.5% 212 D42.4%
Total 300 100% 200 100% 500 100%

DIN Large Small Total
throws% throws% throws%
Flat 44 14.7% 20 P10.0% 64 P12.8%
Twisted 41 13.7% 22 11.0% 63 P12.6%
Concave 108 36.0% 89 Lcx44.5% 197 39.4%
300 100%200100%500100%


This study confirms the MacGregor Historic Games hypothesis that the knuckles are more likely to land concave and convex. The overall probabilities are quite consistent, and are about:

Flat: 1 out of 10
Twisted: 1 out of 10
Concave: 4 out of 10
Convex: 4 out of 10

Among Large knuckles only, the probability is better described as about:

Flat: 1 out of 8
Twisted: l out of 8
Concave: 3 out of 8
Convex: 3 out of 8

It seems that the Flat the Small knuckles are a bit more "frisky," that is, deviate more from regular probabilities, and are more affected by throwing conditions.

MacGregor Historic Games knuckles are very effective at delivering even odds between flat and twisted, and between concave and convex. The single exception (out of 18 paired cells) is the proportion of concave to convex in Small knuckles on the DINette surface.

Throwing with more energy or on a bouncier surface may slightly increase the chances of flat/twisted results. However, if the "banking hunch" noted above is valid, this may be the decisive factor. More research would be necessary to quantify this dynamic.

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