Like everything in politics, I think this is a function of what team you cheer for. If your goal was to come up with an excuse to invade Iraq, that person was an excellent choice. If you’re on the other team, what a clusterfuck.

Then you add in a party system and it gets more complicated. Realistically, you don’t get to be the United States Secretary of Defense (twice) if you’re the kind of person who will ignore the will of the party and whoever is President.

Is that number (publicly) known when you hire the person?

If yes, you just evaluate the choice based on that probability (and other things you knew at the time), not on the actual outcome.

Prediction markets are one way to make these kinds numbers known.

No, the likelihood is unknown, but the hiring process includes a model to estimate it. Of a sort.