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As a good pessimist, I frequently wish to humorously convey extremely low probabilities. I'll often use the phrase "a snowball's chance in hell," or a variation of it, to express this:

There's a snowball's chance in hell that my candidate gets elected

There's a snowball's chance in the Sahara that my efforts accomplish anything

And so forth. Are there any common Latin expressions used for such situations? I'm particularly interested in phrases attested in Classical Latin.

I checked Wikipedia's list of Latin phrases but didn't find anything suitable.

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Not being specially familiar with Classical works, I would use something in the line of VG Mt 19:24:

Facilius est camelum per foramen acus transire, quam divitem intrare in regnum Dei

It has been argued that the original Greek text may have used a slightly different image, but the surviving codices and those from which St. Jerome translated seem to put it that way. This means, at least, that the image was not too foreign at some point between the writing of the Septuagint and Jerome's translation. (And certainly become well known later).

Missing a more Classical or so-to-say pure Latin (as in more confidently not influenced by Greek or Hebrew) quote, I'd say facilius est camelum per foramen acus transire, quam [sth] or the inverse, difficilius est (e.g.) studium meum aliquod perficere quam camelum per foramen acus transire

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There are actually some nice options. Some (most actually) of which are more appropriate as a reply for someone who tries to do something impossible, but I decided to list them here as well as they may be suitable (maybe with little adaptation) to some instances of using "snowball's chance in hell" for what difference is there between flying turtle and snowball in hell . All of them are listed in L&S dictionary :

Expression description reference
Agnum lupo eripere velle to wish to rescue a lamb from a wolf Plaut. Poen. 3, 5, 31
Findere caelum aratro to cleave the sky by the plough Ov. Tr. 1, 8, 3
ebur atramento candefacere to whiten ivory with ink. Plaut. Most. 1, 3, 102.
Prius undis flamma miscebitur sooner will fire mingle with water, of any thing impossible Poët. ap. Cic. Phil. 13, 21, 49.'
ex incomprehensibili parvitate arenae funis effici non possit to make a rope of sand (Gr. ἐξἄμμου σχοινίον πλέκειν), Col. 10 praef. § 4 fin
Jungentur jam grypes equis Griffins now shall mate with mares. i.e., the impossible shall happen Verg. E. 8, 27
Imbrem in cribrum gerere to carry rain with sieve. i.e., to attempt an impossibility Plaut. Ps. 1, 1, 100.
Prius pariet locusta lucam bovem A locust will sooner give birth to an elephant. Enn. ap. Varr. L. L. 6, 3.
Mulgere hircos to milk he-goats Verg. E. 3, 91.
Pinguia corticibus sudent electra myricae let tamarisks distil rich amber from their bark. to signify something impossible Verg. E. 8, 54
Terra feret stellas the ground to support the stars Ov. Tr. 1, 8, 3.
Testudo volat Flying turtle Claud. in Eutr. 1, 352.
Nudo detrahere vestimenta to strip the naked (of any thing impossible) Plaut. As. 1, 1, 79.
Jungere vulpes to yoke foxes. for any absurd or impossible undertaking Verg. E. 3, 91.
Vultur profert cornua vulture to grow horns Claud. in Eutr. 1, 352.
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It's notoriously difficult to find a precise classical equivalent for this kind of thing — though some apt examples certainly exist (caelum digito attingit for 'he's over the moon' is one such).

I think you might best translate it fairly simply. The meaning/intention behind sortem habet niviglobi Gehennae should be pretty obvious, after all!

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