Richard Garfield, the creator of Magic: The Gathering, has a talk where he pontificates about luck in games and how they affect the outcomes of these games. One of the examples he uses to show that skill and luck aren't diametrically opposed is "Rando-Chess".
Imagine a game of chess where at the end of the game you roll a die, and if it's a 1, the winner becomes the loser and the loser actually wins. Now, ignore that voice in your head screaming that's unfair for a moment. Does it actually reduce the skill required to play the game? Everything about chess is still applicable: opening gambits, strategies, knowledge of the rules. Having all of that skill still increases your win-rate over time. It didn't make skill useless at all; however, it does moderate skill disparities.
If you have a game that's all skill, if you're equally skilled you'd expect to win 50% of the time. If you were more skilled, you'd expect to win most, if not all, the time. Adding a random roll at the end in the Rando-Chess game means that the weaker player now has a chance at winning, despite being the poorer player.
Now, Rando-Chess wouldn't be very satisfying to play. I'm pretty sure I'd flip a table or two when I lost due to the direct result of the random roll. Instead, most game designers embed the randomness in their games in other ways. Accuracy in TRPGs or tabletop strategy games, so even if you play the most perfect tactical game ever, you can still get hosed by missing that 99% chance to hit shot. Starting positions in games like Civilization, where you may end up with different resources necessitating different strategies. Deck building in games, where even if you've built the deck and know what's in it, you won't get the cards in the order you necessarily want them in.
But here's a twist where that interesting relationship between the two comes into play again: you can often reduce the effects of randomness by applying skill.
Here are some prime examples.
In a game like Magic: The Gathering or Hearthstone, when building your deck, you want to make sure said deck is as relatively focused as possible. When you draw your next card, you want to increase the chances that said card will be applicable to your overall strategy. This is also often why Card Advantage is very, very important to these kinds of games: because you're cycling through your deck faster, there's a higher chance you'll get the cards that you want, thus reducing the effect of randomness.
|Purple has 5, 6, 9, and 10, whereas Red has 3, 4, 6, 8, 9, 10, and 11. Despite Purple having more points than Red, Red is arguably in the stronger long term position because they're less at the mercy of luck.|
In an MMORPG with trilogy-based real-time combat, like World of Warcraft, the things that usually kill tanks are unpredictable spikes of damage. As a tank or a healer, the best thing you could ever do for your tank's survival is the reduce the effects of randomness as much as possible and ensure the rate of incoming damage is as smooth as possible. Things like Active Mitigation and external cooldowns such as Hand of Sacrifice allow the tanks and healers to have some control over the variability of incoming damage.
|Spreading out is a pretty standard response to "bosses use targeted AoE abilities"|
|Our left-most player gets hit by a circle of doom. Of course, she can just walk out of it here, but since nobody else was around, the potential for damage is reduced.|
|Our left-most player gets boxed in by someone else moving nearby them. If either player had more awareness of their surroundings (a skill), they could have prevented trapping the left-most player.|
On the other side of the equation, you have games that are entirely luck: Chutes and Ladders (or Snakes and Ladders depending on where you live) has precisely zero decisions and precisely zero factors that are influenced by the individual.
If you made a game where the point was to kick a soccer ball the furthest, you have both physical skills (such as strength and accuracy), as well as mental skills (which way is the prevailing wind headed? What spin should I put on the ball?), so while at first blush it might not seem like there's any "skill" involved because it's a feat of physical prowess, there are definitely decisions occurring that could make or break a win even if the players aren't physically equally capable. Basically, a skilled player would use the wind to their advantage. An unskilled player would say they lost because they were unlucky due to the wind. To be fair, however, a strong gust of wind might actually alter the outcome of the match.
Randomness is a tool like any other in a designer's kit. It can be used to muddy the skill disparities between players, or to ensure that players don't get stuck in a rut where the exactly same strategy applies every single time. Players can fight randomness by applying skill, but ultimately they will likely never overcome it entirely (it's possible, if unlikely, for that 11 to be rolled in Settlers over and over again and it's the only number you don't have), so the skill muddying effects can still apply.