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 (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.

Unlike most binary lots — which are typically cast multiply affording a variety of possible outcomes — coins are most often cast (flipped or spun) 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 ('Head or Ship'), 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 ('Flower or Saint'), Spaniards played Castile or Leon, Germans played Wappen oder Schrift ('Weapon or Writing'), and the French played Croix ou Pile ('Cross or Reverse').

Whereas most of these terms describe the images stamped on both sides, both the earlier English Cross and Pile (equivalent to the French, above) 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 (although the tallying of Achillea alpina (yarrow) stalks is the older method). 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 (stamped with writing), 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 (say, 4 quarters, 4 nickels, and 4 pennies), so that a single cast is equivalent to 3 differentiated casts of 4 undifferentiated lots, yielding 1 of 125 possible outcomes (=(4+1)³).