"Hvad er chancen?"
Translation:What is the chance?
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Is this supposed to mean more like "What are the odds/probabilites?" This sentence in English doesn't mean much to me
Yes, that's what it means. This sentence is often said when there is a very low probability of something happening.
"Jeg bliver professionel fodboldspiller en dag!"
"(Hvad er) chancen.."
EDIT: It can also be used when something very unlikely happens:
"Jeg har vundet en rejse til Mallorca! Hvad er chancen?!"
Perhaps it just depends on where you are. To me (north GB) this is fine, although the plural is the normal way to say this: "what are the chances". And it's more like an exclamation of disbelief, rather than an actual question.
A quick search tells me that "Chance" is from French chance meaning "luck" (dice roll/fall) and further down the line, if I read it right, stems from latin cadentia - "to fall" (https://ordnet.dk/ddo/ordbog?query=chance / https://www.etymonline.com/word/chance).
Etymology aside, the word has been in use in Denmark for at least a hundred years (https://ordnet.dk/ods/ordbog?query=chance&tab=for) so saying it is American and shouldn't be used in the Danish course is really missing the mark!
On top of this you can't replace "chance" with "lejlighed", as "chance" is used here with the definition of:
1.b statistisk sandsynlighed for at en bestemt begivenhed indtræffer
(Statistical probability for a certain event happening)
For "lejlighed" to be a synonym it would need to be used in the meaning of the English "opportunity" where the definition is:
1.a mulighed for at nogen kan vise sine evner, eller for at noget kan lykkes
(Opportunity for someone to show their skills or for something to succeed)
If you wrote "Hvad er lejligheden?" I would think you're asking "What is the apartment/flat?".