Binary lot
A binary lot is an object that, when cast, comes to rest with 1 of 2 distinct faces uppermost. These can range from precisely machined objects like modern coins which produce balanced results, to naturally occurring objects like cowrie shells which may produce a range of unbalanced results depending upon the species, individual, and even circumstances of the cast.
Binary lots may be used for divination, impartial decision-making, gambling, and game playing, the boundaries of which can be quite blurred. They may be cast singly, yielding a single binary outcome, but often they are cast multiply, several in a single cast, yielding a range of possible outcomes.
Coins
Unlike most binary lots — which are typically cast multiply affording a variety of possible outcomes — coins are most often cast singly, resulting in a simple yes/no, win/lose outcome. Both the lot and its outcome are binary. Further, a coin's two sides are very nearly symmetrical, so that they can each be expected to appear reasonably close to 50% of the time, unlike cowries, half-round staves, and some other forms of binary lots.The coin flipping game now known as Heads or Tails is ancient, going back at least to classical Greece, where Aristophanes knew it as Artiasmos, and classical Rome, where it was known as Caput aut Navis, the two images on either side of some Roman coins. In the medieval period, various nations stamped various images on their coins, so that Italians played Fiori o Santi, Spaniards played Castile or Leon, Germans played Wappen oder Schrift, and the French played Croix ou Pile.
Whereas most of these terms describe the images stamped on both sides, both the earlier English Cross and Pile and the current English Heads or Tails describe only one side. Pile does not describe what is pictured: it merely indicates 'the reverse side'; likewise Tail indicates 'the side opposite the head'.
For centuries, coin tosses have served both as complete games, and as preliminaries to actions in other games: as early as the 1660s Francis Willughby notes Cross & Pile being played by children as an independent game, but also cases in which Cross & Pile is used to determine who takes a turn first in other games.
Coins are commonly used in I Ching divination. The usual method involves casting three coins to generate each of the six lines of a hexagram. Historically, Chinese coins had only one marked side, and in this procedure it is regarded as yin and given a numerical value of 2, while the unmarked reverse is yang and given a value of 3. The sum of the values of the three cast coins will be between 6 and 9; an even sum means one of the six lines of the hexagram is yin, while odd means yang, with equal probabilities. The cast simultaneously gives a second binary result with unequal probabilities: The sums 7 and 8 mean the line is "young", where as the less likely sums 6 and 9 mean the line is "old" and about to change to its opposite.
The oracular text Ling Ch'i Ching is consulted using 12 wooden disks, strictly, Chinese Chess pieces made from a lightning-struck tree; unsurprisingly, other congruent objects such as home-made disks, wooden checkers, and coins are normally substituted. The 12 disks comprise 4 each of 3 types, so that a single cast is equivalent to 3 differentiated casts of 4 undifferentiated lots, yielding 1 of 125 possible outcomes.
Staves
Staves, lengths of wood typically semicircular in section, are found in many regions and time periods, being used as randomizers in many Native American board games, in the ancient Egyptian Senet as well as the modern Egyptian Tâb, in the ancient Chinese Liubo, and the ancient — and still current — Korean Yunnori. They are easy to make, usually being formed simply by splitting a stick in half lengthwise, though additional finishing or decoration is often applied.The majority of games documented use 3 or 4 staves, though H. J. R. Murray notes games requiring as many as 8. Liubo in fact means '6 rods', which is the number of staves employed in the game.
Cowries
The shells of cowries, sea snails of the family Cypraeidae, often function as lots. Their durable shell is rounded on one side. The other side features a long narrow aperture running from end to end, which the animal may emerge from and withdraw into.Various species of cowrie are used as dice for a variety of board games in India, perhaps most prominently in the traditional Indian game of Pachisi. Here, either 6 or 7 cowries are cast simultaneously, resulting in a single move value, depending upon the number landing mouth up.
In owo mȩrindinlogun, a form of Yoruba divination, 16 cowries are cast, yielding 1 of 17 possible outcomes, each of which is "associated with memorized verses which contain myths and folktales that aid in their interpretation".
Other binary lots
Any object that may be cast to land distinctly on 1 of 2 sides may function as a binary lot.North American
In Games of the North American Indians, Stewart Culin provides descriptions and engravings of over 200 sets of binary lots. The majority are half-round staves, but other lots are fashioned from bone, stone, nut shell, fruit stone, corn kernel, mollusk shell, woodchuck and beaver tooth, claw, brass, and china, as well as wooden lots worked to shapes other than the typical half-round stave.Urim and Thummim
The Biblical Urim and Thummim might have been binary lots, but their form and function remain unclear.Divination tablets
Divinatory use of binary lots in the form "four small rectangular or triangular tablets made out of wood, bone or ivory" is widespread in Southern Africa, likely originating with the Shona people some time before 1561. These are flat, or slightly lenticular in section. They are cast multiply, but unlike many sets of binary lots, they are each individually marked; thus these 4 tablets yield 16 possible outcomes, not 5.Divination chains
Several West African divinatory traditions use divining chains featuring multiples of 4 ordered binary lots, in the form of half seed pods or half mango seeds, but also pieces of calabash, metal, or other objects. The most prominent is the Ifa divination of the Yoruba people, using an Opele featuring 8 lots, most commonly the pear-shaped half seed pods of Schrebera trichoclada. Although the lots are visually similar, they are differentiated by position, being fixed to the chain and the chain being marked with a right and left side; therefore 1 cast of the chain yields 1 of 256 possible outcomes, each of which is associated with memorized verses.Binary lots with more than two faces
Any lot with more than two faces can function as a binary lot if all its faces are grouped into two sets. For example, a cubic die can deliver odds 1:1 like a fair coin if three faces are marked yes and the other three are marked no, or, with a die with normal pip markings, if one only observes whether there is a pip at the center or not. The dice game Bell and Hammer requires 8 cubic dice, each blank on five faces and featuring only a single marked face, each die thus delivering odds 1:5.Some sets of the Royal Game of Ur, dating from the mid–3rd millennium BCE, include roughly regular tetrahedral dice with 2 vertices marked, and 2 vertices unmarked.
Outcomes and probability
Outcomes
When a binary lot is cast singly it yields a single binary outcome. But more often they are cast multiply, several in a single cast, yielding a range of possible outcomes.When the lots are undifferentiated, then n lots produce n+1 possible outcomes: thus, casting 4 staves yields 1 out of 5 possible outcomes. These outcomes are defined by the number of marked faces uppermost, but the value of these outcomes may differ from the simple count of marked faces. For example, in the modern Egyptian board game Tâb, the following schedule is used:
| Marked faces | 0 | 1 | 2 | 3 | 4 |
| Move values | 6 | 1 | 2 | 3 | 4 |
| Extra throw | Yes | Yes | No | No | Yes |
This schedule is typical of most board games using multiple-binary casts in that: 1) the move values are based on, but modified from, the simple count of marked faces, and 2) it is the more extreme counts that are bumped up in value.
When the lots are differentiated, then n lots produce 2 possible outcomes: thus casting 4 distinct Hakata divination tablets yields 1 out of 16 possible outcomes.
These methods are not always strictly exclusive. Several Native American board games make use of 3 staves, only 1 of which is differentiated, resulting in 6 possible outcomes — midway between the 4 if undifferentiated, and the 8 if fully differentiated. The most common coin-based method of I Ching divination begins with a cast of 3 undifferentiated coins, but utilizes 6 casts to produce a complete hexagram, representing 1 of 4096 possible outcomes.
Probability
Even odds
Beginning with the assumption that the lot is equally likely to land with either of its faces uppermost, the odds of a single toss are proverbially familiar: 50:50. But as undifferentiated lots are added to a single cast, the odds become uneven. The simplest case is of 2 lots, marked 0 and 1:| Lot A | Lot B | Outcome |
| 1 | 1 | 2 |
| 1 | 0 | 1 |
| 0 | 1 | 1 |
| 0 | 0 | 0 |
Here, there are 4 possible casts, but these yield only 3 outcomes, which have unequal odds, higher for the central outcome and lower for the extreme outcomes: 2 = 25% and 1 = 50% and 0 = 25%. This pattern holds for all casts of undifferentiated binary lots, as shown below:
The graphical flattening can be deceptive: using 2 lots, the central outcome is 2 times as likely as the extreme outcome. But using 8 lots, the central outcome is 70 times more likely than the extreme outcome. Using cubic dice flattens this curve somewhat, making the odds more even, as shown below:
David Parlett notes: "Cubes have always tended to oust binaries where both are known, probably because they are more convenient, but perhaps also because they bring the rarer numbers more frequently into play."