Is there a word or phrase that means 'works but not for the reason we expect it to'?

I'm looking for a word or phrase that means that something works or functions, but not for the reason that people think or claim it works.

Examples

• John went outside in the rain with an opened umbrella above his head, thinking that pointing a stick at the heavens would let the gods know that he respects them and have them protect him from getting wet.
• Homeopathy: People believe that like cures like - that a dilute version of an irritant could cure the irritation. It works, but not for that reason: it's just a well-administered placebo, and placebos are remarkably effective.

Related concept:

• Cargo cult: I'm looking for a word / phrase / idiom that could describe what would happen if a cargo cult actually achieved what the participants set out to achieve.

Example sentence:

As requested, this is an example sentence which would show the use of the concept as a single word (please note that I'm also looking for phrases that would be suitable, so this is just per the requirements of the 'single-word-requests' tag.

John went outside in the rain with an opened umbrella above his head, thinking that pointing a stick at the heavens would let the gods know that he respects them and have them protect him from getting wet. However, this was merely a __________, because it was actually the fabric of the umbrella that protected him from the downpour.

Solution 1:

As I mentioned in a comment under the question, what's being described is the fallacy of false cause.

From "your logical fallacy is":

You presumed that a real or perceived relationship between things means that one is the cause of the other.

Many people confuse correlation (things happening together or in sequence) for causation (that one thing actually causes the other to happen). Sometimes correlation is coincidental, or it may be attributable to a common cause.

Example: Pointing to a fancy chart, Roger shows how temperatures have been rising over the past few centuries, whilst at the same time the numbers of pirates have been decreasing; thus pirates cool the world and global warming is a hoax.

A common expression that goes along with this is:

Correlation does not mean causation.

Solution 2:

I hesitate to flat-out answer No, but I don't know of such a word. Maybe this could help - I recall these two phrases that describe or imply someone being right for the wrong reason (actually, being right from random chance, but it would be wrong to attribute the reason to something else): "Even a broken clock is right twice a day." "Even a blind squirrel sometimes finds a nut."

Solution 3:

Hmm. Serendipity (serendipitous) is close, but implies accident or coincidence more than false belief.