You can map out 50 shots from that gun, and pinpoint the average middle, and how much individual shots diverge from that middle. Then you take the average of another 50 shots and notice the recoil has pushed the average middle slightly upwards. So you make a model predicting the average middle in another 50 shots, accounting for the degree change from the recoil meaning a further distance than the previous to average middles.
Then comes along a guy asking you to predict just 1 shot, the next one, not an average. Which of course is much less accurate. That is what you're suggesting.
So to go back to your excellent analogy, I am certainly asking for the average middle (the first moment) and not a point prediction.
ps: you should write more, we need better scientific explanations!
Use it to predict winter in NE 1 year out? I would posit that if they can predict average temps 400 years out then average temps 1 year out should have higher confidence? These predictions can be used to trade heating oil futures and make tens of billions. I do know that energy firms employ many climate modelers but unsure of the accuracy of their forecast s even 1 year out.