All good questions and a nice chance for an educational thread.
1. Nobody knows how a die skips through inspection with hub doubling on it. It's a mistake - plain and simple. There are no records to show how it happens, because if they knew it was happening we would never see the result. It would be interesting to see some of the dies that did get caught and never minted coins.
2. The number of coins struck by a die usually runs 250,000 to a million. Some of the doubled dies were caught before these numbers were reached, but in some cases the die runs its whole expected life and isn't caught. Case in point is the 1995
DDO. There are an estimated 750,000 to a million pieces out there. People are still finding them in change.
3. The "odds" of
Machine Doubling occurring? Pretty high, I would think, especially in certain years because of press problems. I can tell you this much - I search through over a hundred thousand Lincoln cents per year. The number of machine doubled coins I see to genuine doubled dies is hundreds to one.
Machine Doubling is FAR more common than doubled dies.
Here's your main reason why the doubled dies bring money and
Machine Doubling doesn't...doubled dies have design doubling on the die. Each die is supposed to be inspected for such things before being hung on the press. As we know, some escape this inspection undetected, but the vast majority are caught and destroyed. With
Machine Doubling, a die can come loose in the die and strike hundreds of thousands of coins without being noticed, because there is no microscopic inspection process of each individual coin after they leave the press. A spot check is performed for quality somewhere in the run, but if that one coin is good, all are allowed to pass into circulation. If the die loosens after the inspection, the resulting machine doubled coins aren't even noticed.
Second reason why doubled dies bring the money and
Machine Doubling doesn't...doubled dies are catalogable by die. The die continues to make coins that have the same doubling until it is retired. Given this (and the reason why doubled dies are "die varieties" and not "errors"), a "set" of doubled dies can be collected by die number as cataloged, and some sense of completion can be obtained by collectors.
Machine Doubling is extremely hit or miss. Two coins struck by the same die; one could have
Machine Doubling and the other not simply because the die worked itself loose after the normal coin was struck. This could have happened to half of the dies hung in a given year. We never know. With the uncertainty, there's no way to place a value on them.
Yet another reason? Die varieties have value because since they are catalogable and attributable die for die, we can judge their rarity.
Machine Doubling is not considered to be a die variety, they are very minor errors (but even this is debatable since they meet mint tolerances for quality). Generally errors have to be something dramatic to have value. Error dealers don't mess with
Machine Doubling because it usually requires magnification to see, happens frequently, fits in a normal roll thus makes it through all the machines at the mint into circulation, and doesn't have that catchy flair that a multiple struck off center with an indent does. It's these dramatic 'differences' in the errors that sell, not the 20X visible minor doubling of a couple of digits in the date.
Because the market centers its attention on die varieties (doubled dies and mintmark varieties) and more dramatic errors (generally those that make an obvious difference to the coin),
Machine Doubling is considered 'non-collectible' doubling and isn't picked up as a commodity in the market. There is simply no general interest in it. This is why I tell people who really want to know what to collect to not waste their time collecting and saving
Machine Doubling if they ever want to get their money's worth for what they are collecting.
Many times I have seen collections that took years to assemble that are full of
Machine Doubling. They think they've hit the jackpot with their binders full of these examples, but when it comes time to sell, nobody will give them anything over the value of the normal coins for them. They end up with binders full of worthless coins they thought they could get thousands for. I see it time and time again, and it's the result of a lack of educating themselves as to what they should have been looking for all along. I do anything I can to steer collectors in the right direction and keep them from picking out and hoarding coins that have no numismatic value.
Machine Doubling adds no numismatic value to a coin - that's a fact. Collect it if you think it's pretty, but don't expect to get anything for it. I don't dictate that, the market in general does.