Binary lot

— a common method of casting a binary lot A binary lot is an object that, when cast, comes to rest with one of two distinct faces uppermost. These can range from precisely machined objects like modern coins which produce balanced results (each side coming up half the time over many casts), 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 (as David Parlett suggests) can be quite blurred. They may be cast singly, yielding a single binary outcome (yes/no, win/lose, etc.), but often they are cast multiply, several in a single cast, yielding a range of possible outcomes.

— a common method of casting a binary lot A binary lot is an object that, when cast, comes to rest with one of two distinct faces uppermost. These can range from precisely machined objects like modern coins which produce balanced results (each side coming up half the time over many casts), 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 (as David Parlett suggests) can be quite blurred. They may be cast singly, yielding a single binary outcome (yes/no, win/lose, etc.), but often they are cast multiply, several in a single cast, yielding a range of possible outcomes. 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 four each of three types (say, four quarters, four nickels, and four pennies), so that a single cast is equivalent to three differentiated casts of four undifferentiated lots, yielding one of 125 possible outcomes (=(4+1) 3 ). Several West African divinatory traditions use divining chains featuring multiples of four ordered binary lots (often 8 or 16), 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 (divining chain) featuring eight lots, most commonly the pear-shaped half seed pods of Schrebera trichoclada (locally called Opele, whence the chain gets its name). 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 one cast of the chain yields one of 256 (=2 ) possible outcomes, each of which is associated with memorized verses. When the lots are undifferentiated, then n lots produce n+1 possible outcomes: thus, casting four staves yields one out of 5 (=4+1) 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: 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 (which are statistically rarer, see below) that are bumped up in value. When the lots are differentiated, then n lots produce 2 possible outcomes: thus casting four distinct Hakata divination tablets yields one out of 16 (=2 ) possible outcomes. These methods are not always strictly exclusive. Several Native American board games make use of three staves, only 1 of which is differentiated, resulting in six possible outcomes — midway between the four if undifferentiated, and the eight if fully differentiated. The most common coin-based method of I Ching divination begins with a cast of 3 undifferentiated coins (4 possible results), but utilizes 6 casts (differentiated by order) to produce a complete hexagram, representing one of 4096 (=(3+1) ) possible outcomes. Here, there are four possible casts, but these yield only three outcomes, which have unequal odds, higher for the central outcome(s) 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 two lots, the central (most common) outcome is twice as likely as the extreme outcome. But using 8 lots, the central (most common) outcome is 70 times more likely than the extreme outcome. Using cubic dice (or any dice with more than 2 faces) 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." (G = grace, a useful bonus point; + = roll again.) If the four extreme outcomes are collectively considered the "good" ones, then uneven odds actually increase the chances of a "good" cast (where the bulk of the gain is from 1 mouth up):

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