Etymology
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chance (v.)

late 14c., "to come about, to happen," from chance (n.). Meaning "to risk, take the chances of" is attested from 1859. Related: Chanced; chancing.

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chance (adj.)

"resulting or due to chance; casual, unexpected," 1670s, from chance (n.).

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chance (n.)

c. 1300, "something that takes place, what happens, an occurrence" (good or bad, but more often bad), especially one that is unexpected, unforeseen, or beyond human control, also "one's luck, lot, or fortune," good or bad, in a positive sense "opportunity, favorable contingency;" also "contingent or unexpected event, something that may or may not come about or be realized," from Old French cheance "accident, chance, fortune, luck, situation, the falling of dice" (12c., Modern French chance), from Vulgar Latin *cadentia "that which falls out," a term used in dice, from neuter plural of Latin cadens, present participle of cadere "to fall," from PIE root *kad- "to fall."

In English frequently in plural, chances. The word's notions of "opportunity" and "randomness" are as old as the record of it in English and now all but crowd out its original notion of "mere occurrence." Meaning "fortuity, absence of any cause why an event should happen or turn out as it does, variability viewed as a real agent" is from c. 1400.

Chance is equivalent to the mathematical concept of probability, which is a precisely measurable factor enabling the accurate prediction of average outcomes over long runs of random events — the longer the run, the more accurate the predictions. Luck is at best a platitude and at worst a superstition. [David Partlett, "A History of Card Games"]

Main chance "probability that offers greatest advantage," hence "thing of most importance" is from 1570s. Mathematical sense "probability, likelihood of a certain outcome" is from 1778, hence the odds-making sense "balanced probability of gain or loss." To stand a chance (or not) is from 1796. To take (one's) chances "accept what happens" (early 14c.) is from the old, neutral sense; to take a chance/take chances is originally (by 1814) "participate in a raffle or lottery or game;" extended sense of "take a risk" is by 1826.

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off-chance (n.)

"a remote chance," 1861, from off (prep.) + chance (n.).

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bechance (v.)
"to happen, chance," 1520s, from be- + chance (v.). Related: Bechanced; bechancing.
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mischance (v.)

"to happen wrongly or unfortunately," 1540s, from mis- (1) + chance (v.). Related: Mischanced; mischancing.

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chancy (adj.)

1510s, "lucky, foreboding good fortune," from chance (n.) + -y (2). Meaning "uncertain, subject to risk" is recorded from 1860.

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perchance (adv.)

"perhaps, possibly, maybe," mid-14c., parchaunce, from Old French par cheance, literally "by chance." With Latin per substituted c. 1400 for French cognate par. See per + chance (n.).

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mischance (n.)

"mishap, ill-luck, disaster," c. 1300, from Old French mescheance "misfortune, mishap, accident; wickedness, malice," from Vulgar Latin *minuscadentiam; see mis- (2) + chance (n.). Now usually "bad luck;" formerly much stronger: "calamity, disaster, affliction."

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cadence (n.)

late 14c., "flow of rhythm in prose or verse," from French cadence, from Old Italian cadenza "conclusion of a movement in music," literally "a falling," from Vulgar Latin *cadentia, from neuter plural of Latin cadens, present participle of cadere "to fall" (from PIE root *kad- "to fall"). A doublet of chance (n.).

The notion is of a "fall" in the voice in reading aloud or speaking, as at the end of a sentence, also the rising and falling in modulation of tones in reciting. Later (1590s) extended to music, in reference to a sequence of chords expressing conclusion at the end of a phrase and resolving to the key in which the piece was written. Also the measure or beat of any rhythmic movement (c. 1600). In 16c., sometimes used literally for "an act of falling." Related: Cadential.

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