Randomness is rarely chaos. In games, randomness is a pick from a pool with known and computable probabilities.
Roughly 1/6 of the rolls in Settlers of Catan will be 7s that activate the robber. Roughly 5/36 of the rolls will be 6s and another 5/36 will be 8s, which is why they are the most sought numbers and why there is so much groaning when there are 5 6s in a row or when there are 0 8s for 15 minutes. Some players think that still leaves too much to chance, so you can buy dice decks, giving you the exact probability distribution over 36 turns rather than knowing it works out in the long run (but maybe not over the course of this game). Strategic play in Settlers involves both getting the best numbers possible and diversifying over numbers and tile types, to survive the swings of fortune.
Deck-building games are substantially about probability manipulation. In Dominion, you start with 10 cards, so you always know exactly what your second hand will be based on your first hand, and you can predict turns three and four based on the (usually) 12 cards you own after those first two turns. From there, the deck-building mechanic is about increasing the number of good hands you have by getting more good cards, fewer bad cards, or just more cards, or perhaps by exploiting higher averages or variability. There are strategies that depend on knowing what is left in your deck based on what you have played so far.
I have been pondering Dungeon Roll. When you roll a lot of dice, the possible extreme swings in luck are large, but you are increasingly likely to be around the center of the distribution. You may have a random pick of a dozen heroes with a random roll of seven white dice against a random roll of foes on each level of the dungeon, but that averages out to reaching level 5 safely, plus or minus one, plus or minus a dragon. You probably end a turn with 4 to 6 experience points and 2 to 4 treasures.
Because you get only three turns, conservative play is encouraged, which is an odd thing for a “push your luck” game. If you go one level deeper, you get 1 or 2 more points for succeeding and lose 4 or 5 for failing. That is a big asymmetry, and you lose the game if you lose one round unless it encourages your opponents to also push their luck a little further (by why would they with a 5-point lead in a game where 30 is a high score?). I have yet to see anyone push his/her luck and fail, because you can predict your odds (two white dice left, enemy rolls six black dice) and your potential benefits (marginal).
On the other hand, there is some variation, particularly within one turn, and that can lead to unfortunate swings you cannot do anything about. “Leveling up” is worth about 1 more point in each of your latter two turns, so if a bad roll means you do not get the 5xp to level up your first turn, you are down even further against your opponents. Sometimes the dice make your special abilities useless. In my first game, I was the knight, and who turns scrolls (low value) into champions (high value); I rolled 0 scrolls on my last two turns. You can expect to roll 0 scrolls in 28% of turns, which means that 63% of games will have at least one turn where that ability does nothing, and it will not be used at all in 2% of knights’ games. In three turns, randomness can dominate long run odds.
: Zubon