The Collaborative International Dictionary of English v.0.48:

May \May\ (m[=a]), v. [imp. Might (m[imac]t)] [AS. pres. maeg
   I am able, pret. meahte, mihte; akin to D. mogen, G.
   m["o]gen, OHG. mugan, magan, Icel. mega, Goth. magan, Russ.
   moche. [root]103. Cf. Dismay, Main strength, Might. The
   old imp. mought is obsolete, except as a provincial word.]
   An auxiliary verb qualifying the meaning of another verb, by
   expressing:
   (a) Ability, competency, or possibility; -- now oftener
       expressed by can.
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             How may a man, said he, with idle speech,
             Be won to spoil the castle of his health!
                                                  --Spenser.
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             For what he [the king] may do is of two kinds; what
             he may do as just, and what he may do as possible.
                                                  --Bacon.
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             For of all sad words of tongue or pen
             The saddest are these: "It might have been."
                                                  --Whittier.
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   (b) Liberty; permission; allowance.
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             Thou mayst be no longer steward.     --Luke xvi. 2.
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   (c) Contingency or liability; possibility or probability.
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             Though what he learns he speaks, and may advance
             Some general maxims, or be right by chance. --Pope.
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   (d) Modesty, courtesy, or concession, or a desire to soften a
       question or remark.
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             How old may Phillis be, you ask.     --Prior.
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   (e) Desire or wish, as in prayer, imprecation, benediction,
       and the like. "May you live happily." --Dryden.
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   May be, & It may be, are used as equivalent to
      possibly, perhaps, maybe, by chance,
      peradventure. See 1st Maybe.
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WordNet (r) 3.0 (2006):

by chance
    adv 1: through chance, "To sleep, perchance to dream.." [syn:
           perchance, by chance]
    2: by accident; "betrayed by a word haply overheard" [syn:
       haply, by chance, by luck]
    3: without advance planning; "they met accidentally" [syn: by
       chance, accidentally, circumstantially, unexpectedly]
       [ant: advisedly, by choice, by design, deliberately,
       designedly, intentionally, on purpose, purposely]

Moby Thesaurus II by Grady Ward, 1.0:

38 Moby Thesaurus words for "by chance":
   accidentally, as it chanced, at haphazard, at intervals, at random,
   by a fluke, by accident, by fits, by good fortune, by hazard,
   by snatches, capriciously, carelessly, casually, desultorily,
   erratically, fitfully, for some reason, fortuitously, haphazardly,
   hit or miss, in snatches, in some way, in spots, incidentally,
   indiscriminately, irregularly, perchance, promiscuously, randomly,
   sloppily, somehow, somehow or other, spasmodically, sporadically,
   unmethodically, unpredictably, unsystematically