I think the most important part of randomness in a game is that it is understandable and the players are able to influence it.
If they can’t understand the randomness, then there is no game for them. The players find things happening without rhyme or reason and can never predict whether an action will have a positive or negative effect.
It’s possible to play such a game, and even enjoy it in a limited way, but there is very little a player can do to learn about the game. They can’t fine-tune their skill at the game and try to improve at it. A large part of the fun in a game is learning about how to play it and developing new ideas and strategies to solve the set of problems that the game puts in front of the players.
So a player needs to be able to tell, at least approximately, what the possible gains or losses of a course of action are. They also need some idea of how likely each of the possible results are. But this on it’s own isn’t good enough. In Monopoly, to take a well-known example, you can know the exact distribution of movement distances based on the dice roll and the exact danger of each possible result. But you never have control over it. On every turn you are victim to chance. You don’t choose your risk, you’re forced into risk.
It is much more enjoyable to be able to take risks by choice and live with the consequences of that choice, because at least then those consequences are yours. You’re not having your hand forced by chance, you made the informed decision to take that risk.
Seph North 
That is true enough about movement in “Monopoly” as far as it goes, all though of course that is not the core of the game (rather underrated these days IMHO); there is a bit you can do to ameliorate the effects of movement, but not much. A better example would be “Snakes and Ladders”, which is completely pointless: you might as well roll a few dice each and the highest score wins. Would you regard “To Court the King” as a good example of choosing your risk? If not, do you have anything else in mind?
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Yes, it’s not the only factor of monopoly, but with few exceptions the dice determine exactly what opportunities you will have throughout the game. It is possible to get good at picking the best options for the opportunities you get, the problem is that it is *very* possible to roll the best opportunities and win without a call for a lot of skill.
Agreed on snakes and ladders. It is literally a random win, with only the illusion that there is some sort of competition. The only thing that influence the win is going first.
I would agree that To Court the King is a good example of choosing risks. Many (most?) wargames also have a big risk-selection factor (eg putting more forces into a battle to boost your odds, or reducing your troops in that battle to do more elsewhere).
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