When you have eliminated the impossible....

…whatever remains, however*** improbable***, must be the truth.

Is this true?

You would second guess Sherlock Holmes himself?

More seriously, yes it is trivially true. Probability has no real bearing on an actual event - no matter how much the chances are against a particular thing happening, if it occurred then it occurred. The sticking point of that somewhat simplistic phrase is that many people have no real grasp of what is impossible or just improbable.
Eliminating the impossible still leaves a very great number of increaingly improbable alternatives, not just one single explanation. If you hear hoofbeats on the street in London, you are justified in ruling out it being a unicorn, but that does not mean it has to be a zebra.

The problem is first defining “impossible” for that particular situation. If you can do that, then once you eliminate the impossible all that remains are the known and unknown probabilities.

In mathematics, this is a legitimate proof technique: break a situation down into the possible cases, and logically eliminate all but one.

In real life, the technique is of limited usefulness, because you don’t always know what all the alternatives are: there may well be possibilities you hadn’t thought of.

Well, technically no. By the literal reading of that phrase, it means that every conceivable happenstance -save for the impossible- occurred.

This is a very destructive concept when used in the real world. There is rarely, if ever, a way to eliminate the impossible. You’ll hear “We’ve done a thorough investigation, and the evidence clearly shows John Doe to be the culprit”. What it means is “We glanced at some other suspects, they didn’t confess, so the guy we don’t like must have done it”.

True. Suppose you’re investigating a murder and you determine that over six billion people would have been physically unable to have committed the crime. You’re making progress but you still have fifty million people who could have committed the murder on your suspect list.