Er...independent trials has a mathematical definition. It is P(AUB) = P(A)*P(B).
Yes. For this to be satisfied for SVB and Fedor playing bar-box 8-ball, they's need to be flipping a coin for each break. That would make the events (games) independent. For two games as independent events you have, from SVB's perspective
P(WW) = 0.5 * 0.5 = 0.25
P(LL) = 0.5 * 0.5 = 0.25
P(WL) = 0.5 * 0.5 = 0.25
P(LW) = 0.5 * 0.5 = 0.25
As a result you have 0-2 a quarter of the time, 2-0 a quarter of the time, and 1-1 half the time
For winner breaks, alternate breaks, and loser breaks, the games are not independent events for these runout players. For winner breaks and loser breaks you get almost all 0-2 and 2-0 scores. For alternate breaks you get almost all 1-1 scores.
[...] when alternating breaks as in this example, the probability of the person, say Shane, who won the lag winning the third game is the same regardless of the previous games.
I see what you are saying that the probability of winning game 3 doesn't depend on the OUTCOME of game 1, But game 1 and game 3 are correlated here. The likelihood is high you either win both or lose both
[...]
The final piece there is that you said that a race to 7 on a barbox might be considered the equivalent of a race to 5 on a 9' in terms of "trials." Well, let's say we have a weaker player vs a stronger player, say me vs SVB. Break format irrelevant. What is more likely--I beat SVB in a single rack or in a race to three? How about a single rack vs a race to 5? How about a single rack vs a race to 7? Or to 120? The longer the race, the more likely it is that SVB wins. Therefore, the chance of my winning a race to 5 is better than the chance of my winning a race to 7. By your own explanations this means I am more likely to win an equivalent-length race on a 7' table than on a 9', given we are essentially comparing a race to 5 to a race to 7. Again, using your explanation.
Yes, this is exactly my point.
I'm all for mathematical explanations, but analogies and whatnot, while they can be illustrative, are absolutely not proofs. Your analogies and explanations contradict expectations from basic statistics and mathematics.
I am not sure what you are referring to here. But if you find an example, I will happily concede. I am not wedded to any particular analogy or explanation. I want them to be right and helpful. If they are missing one of those, I will abandon them.
