You're talking proof here, and I'm not. We have no debate on whether the prosecution proved the boy was guilty: they didn't. The standard you're challenging, and that I'm defending, is whether the boy probably killed his father.
I can demonstrate that using a wiggle of my ring finger and some mathematical magic on the knife alone. The knife does not prove anything, and is far too weak a demonstration to actually convict someone, especially with no other evidence, but I'm not going for proof here. I'm going for probability. And that knife is just loaded with probability.
There are two possibilities:
1. The boy murdered his father with his knife, and left it at the scene.
2. By coincidence, the boy bought a specific knife several days prior and then promptly lost it on the same day that his father was murdered with either it or an identical one.
The second option has several coincidences that can be assessed for probability.
First, what is the chance that the boy happened to buy the same style of knife that was used to kill his father? Let's say, extremely generously, it's one in ten.
Second, what is the chance that he would lose his knife on that same day? Let's say, still generously, it's one in one hundred.
Under these numbers, with no other evidence, the probability is 1,000 to 1 against this happening.
But on the other hand, what's the probability that the boy would want to kill his father? Well, setting aside all other circumstances except that this guy was murdered, the probability that his own son did it is about 1 in 53, according to FBI homicide statistics.
They're both really unlikely, but the father was killed by someone, and if it wasn't the boy, he lost his knife in a weird coincidence. So to find the probability for each option of these two, divide 53 by 1000, and with the knife alone, the statistical probability is about 94.7% in favor of him being the murderer, just on the sheer audacity of the coincidence.
This doesn't mean 94.7% certain and 5.3% doubt; there's much more doubt than that, and much less certainty. It just means he probably did it. Actual certainty created by this is quite low, and a great deal of doubt is left.
I don't know if he did it. I certainly wouldn't send him to the chair because there was an unlikely coincidence. There's plenty of reasonable doubt there. But that knife alone is enough to justify my statement that he probably did it.