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I was standing at a urinal recently as tried to read the note on the top surface. It was as follows:

　人がいない場合でも水が流れることがあります

Based (mostly) on the words and context I would expect this to mean something like "When there is no person, there will be a flush of water"

I am hung up on the [場合でも」　part. In this question [場合」 is explained as typically being followed by に and/or は. Both of those make sense to me as a typical particle usage for a "temporal noun", but I am not sure what the 「でも」 usage is.

How is 「でも」acting here, or is it 「で＋も」?
Thank you!

I would structurally translate this as:

人がいない場合でも水が流れることがあります。

There are times when the water will flow, even if it is the case where there is nobody (here).

The N+でも construction expresses the same as Vて+も, なadj+でも いadj＋くても, which indicates a 'reverse condition', usually translated to "even if" - i.e. explaining a scenario which is contrary to what you might think. Usually, I suppose, a urinal's flushing feature only activates after use, by using one of those infra-red sensors; this is simply explaining that they may also (surprisingly) activate as a matter of course, without a person needing to use it/be there to trigger it.

場合 is often used in a related way, in "no matter the case" type phrases, with どんな・どの場合でも:

どんな場合でも私は笑っていた。No matter the situation (case), I was smiling.

この規則はどの場合でも当てはまる。This rule applies in all cases.

どんな場合でも皮膚の色で人を判断すべきではない。 (Whatever the case) you should never judge people by the colour of their skin.

でも means "even if/even though/but".It is contrastive, and it's the version of いadj + くても used with nouns and なadj:

寒【さむ】くても、ジャケットを着【き】ないででかけた。Even though it was cold, I left without wearing a jacket.

昨日【きのう】暇【ひま】でも、宿題しなかった。Even though I was free, I didn't do homework yesterday.

So your sentence would mean "water flowing may exist even in the case there is nobody", or put in more natural terms, "water may flow even if there's nobody".

Therefore, it is not stated that the water only flows when there is no one, but that the water flows (in general) and it can flow when there is no one as well. As this might be unexpected, the contrative でも is used.

EDIT: I cheched the answer you reference. 場合 means "case" and it is often used as a conditional, but not always.

There are good answers here, but I'll provide my parsing as well. I take the sentence and peel back the layers:

人がいない場合でも水が流れることがあります。

So then, adding some parentheses to group clauses:

(人がいない場合でも水が流れる)ことがあります。

([人がいない場合]でも[水が流れる])ことがあります。

We can translate ことがあります。 loosely as "the condition exists." or "the event exists." Putting this aside for now, let's translate the inner parts, starting with the right because it's easier:

水が流れる = Water flows

人がいない場合 = The case where no person is around

Now, the でも in between can mean "even if" and attaches to the left clause, so:

人がいない場合でも = Even if it is the case where no person is around

So putting it together:

人がいない場合でも水が流れることがあります。

There exists the event where water will flow even if no person is around.

This sounds stiff in English, so we can relax it a bit:

Water may flow even if no one is around.