You're just pretending you can calculate inherently incalculable probabilities. How do you determine the probability a company will fail under a variety of potential CEOs so that you could even get a mathematically-justifiable rank order? You can't run trials, you can't really extrapolate from past performance, you're subject to all kinds of confounding variables, you can't actually treat 'failure' as '$0 market cap', etc, etc, etc.
The best you could ever hope to do is "CEO candidates with this kind of background tend to have this kind of record when taking over this kind of company", which I'm sure is quite an illuminating sample.
You are correct - the board's estimate is merely the best approximation they can come up with based on incomplete information and a subjective set of priors (yes, "subjective prior" is redundant). So what?
I really don't get the point you are making. Are you telling me that a struggling company shouldn't try to get what they believe is the best possible CEO, provided his pay package is vastly smaller than the variance in possible outcomes for the company?
My point is that it's ridiculous to try to justify golden parachutes for bad CEOs using fake math, and the fact that the market bears these kinds of contracts isn't evidence on its own that they are good.
The fact that most business decisions are made by gut feeling implies that post-hoc pseudo-mathematical rationalization shouldn't be called out as bullshit?
The best you could ever hope to do is "CEO candidates with this kind of background tend to have this kind of record when taking over this kind of company", which I'm sure is quite an illuminating sample.