Suppose X has murdered someone with a knife, and is being tried in a courthouse. Two witnesses step forward and vividly describe the murder. The fingerprints on the knife match X's fingerprints. In fact, even X himself confesses to the crime. How likely is it that X is guilty?
It's easy to construct hypotheses in which X is innocent, but which still fit the evidence. E.g. X has an enemy, Z, who bribes the two witness to give false testimony. Z commits the murder, then plants X's fingerprints on the knife (handwave; assume Z is the type of person who will research and discover methods of transplanting fingerprints). X confesses to the murder which he did not commit because of the plea deal.
Is there any way to prove to Y (a single human) that X has committed the murder, with probability > 0.999999? (Even if Y witnesses the murder, there's a >0.000001 chance that Y was hallucinating, or that the supposed victim is actually an animatronic, etc.)
Having sat on a jury (for a rather dull case of a failed burglary), I concur with this.
Jury confidentiality is taken seriously in the UK, so I can't comment on our deliberations, but the consensus was that it was him wot dunnit. He looked resigned rather than indignant when the verdict was read out, so with that and the evidence I'm as sure as I need to be that we got it right. I couldn't put a number on it, but 0.000001 is way smaller than a reasonable doubt.