Is there an expression that describes looking for something that probably isn't there, or looking in unlikely places for a solution [duplicate]

Solution 1:

A fruitless search (looking for something that isn't or probably isn't there) described humorously is a wild goose chase.

From The Idioms:

• wasting resource working on something that does not exist
• spending time searching for something that is simply impossible to find
• a foolish search for something that cannot be achieved
• a chase for something that's beyond your reach

The convict escaped the police custody on the way to jail. Police is apparently on a wild goose chase after he vanished into woods.

It turns out that my brother took my car keys. I had been on a wild goose chase this whole morning searching them in the entire house.

Solution 2:

How about fool's errand? From Cambridge:

fool's errand: an effort that is unlikely to be successful

From Google:

fool's errand: a task or activity that has no hope of success

The act of looking for something that probably isn't there, or the act of looking for a solution in a place where it is not likely to be found, is a fool's errand. And fool's engaged in tomfoolery are certainly humorous. :-)