Why Randomization
Randomization is to the game designer as spice is to a chef. When done right, it does a lot of good for the game. But done wrong, it can ruin a good game.

I consider it one of the best parts of games. Soccer, poker, trading card games all rely heavily on randomization and good players factor it in.

Randomness is exciting because you don't know what you'll get. It's a shock when you fail, and relief when you succeed. It's exciting when you wait for the results.

Why Not?
It depends entirely on your players. Some sore losers don't take well at all to anything based on luck. Some sore winners also don't like it when their opponent accuses them of winning because of luck.

You should also note that the cost of a streak of bad or good luck is bad for inexperienced players. Good luck streaks will make the player overestimate themselves and think that they're stronger than they are - then complain of bad luck when their luck returns to normal. Bad luck streaks are even worse. Players are hardly ever grateful for good luck streak, but will yell and shout when encountering a bad luck streak.

If you want a 'competition game', it's usually good to minimize randomness. Some of the most competitive games out there, like chess and wargaming are devoid of randomness (but it can be argued the more popular ones like poker and soccer are heavy in them).


CONCEPTS
Random does not mean luck!
Many new players and game designers seem to think that 'random' means success is based entirely on luck. This is far from the truth. If you have a 30% chance of success, it means around 300 out of 1000 attempts will succeed.

A big thing about being random is predicting how your rolls will end up. As poker goes, you play good cards when it's likely to win. Although it's possible to win if you're lucky, a good poker player never bets on bad cards. Similarly, a good RPG player doesn't go into a battle he's unlikely to win.

Yet, randomness forces them to take a degree of risk. This calculated risk is what makes the game exciting!

..but sometimes it is based on luck
The big problem is the order in which you succeed. Randomness is a huge factor in game design. In some games, like Civilization, early success gives you a huge advantage. An early combat win could mean that you'll grab important resources, while a loss could force you to be stagnant for much of the later game. In this case, whoever is lucky early will get the advantage. This is not fair.

Another problem is when attacks are cumulatively powerful. Let's say at full health, you function perfectly well. At half-health, you fight half as well. Now your enemy gets lucky early, he hits you. You fight badly, your chance to hit goes down.

These are two cases where you should strongly avoid or lower the risk of randomness.

Gambler's Fallacy
This deserves a quick mention. Gambler's Fallacy is the idea that if there's a 20% chance of succeeding, you're more than 20% likely to hit if you missed the first time. This idea is wrong! If you miss, you're 20% as likely to hit the next time! If you miss a third time, you're still 20% likely to hit next time. Look it up on Wikipedia if you want proof.

The exception is if you use PRD, explained later. With PRD, it's the exact opposite. If you miss the first few times, you should expect to hit next time.

Probability of failure/success
Let's say you have a 5% chance to succeed at something. That means you have a 95% chance of failing. You fail the first time. What's the chance of failing next time?

Chance of failing is 0.95^N, where N is the number of tries. So, with 3 tries, it's 0.857 chance of fail. Success = 1 - 0.857 = 14.3% chance to succeed at least once in 3 tries.


IMPLEMENTATION
Random(X)+C
This is the simplest concept. Random(10)+10 generates a number from 10 to 19. You will get every number from 10 to 19 evenly. It works, but it won't let you bias your numbers anywhere.

Good for: Very simple random things, like damage from bullets.
Bad for: Anything where the results actually matter

Dice
Dice are the easiest form of random control. Let's say you have two 6-sided dice. You're much more likely to roll 6,7 than 2, 12. You can similarly bias random numbers towards other things if you like. (Random(3)+Random(3)+Random(3))/3 will make a lot of your results close to 1.5.

Good for: Almost everything. That's why games were based on it for centuries. You'll get overall average results.
Bad for: When you want something to be more random.

Multiple successes - random damper
Branching off the dice and probability concept, multiple successes is a great way of dampening randomness. Let's say a weak player has a 10% chance of bashing open a door. He gets very lucky and succeeds. A much stronger player has a 90%, gets unlucky and fails. That's flawed.

You could instead give two checks, requiring them to succeed in both. Thus, the first player will have a 0.1*0.1 = 1% chance of success, while the second would have a 0.9*0.9 = 81% chance of success. Now there's almost no chance that the first player will break it open, but you could still use the same stats.

The flaw is that it's boring. But it's good because you could just have a general 'pick lock' skill, where the easy locks require only one attempt, and the hardest ones may require up to 4, making it impossible for a newbie to pick.

Good for: realistic results, lowering the luck factor
Bad for: Actually being random at all

Pseudo Random Distribution (PRD)
As it's called by Warcraft III, PRD is where the chances of succeeding actually increases the more you fail. I'll not go into the technical details because it's complicated, just the design ones.

The good thing is that this gets rid of almost every flaw I've mentioned. You're bound to succeed sooner or later. And it works exceptionally well with casual players who don't understand the rules.

The problem with this is that you're literally rigging the rules to make the player feel better. Once the player learns that it's not truly random, it gets rid of the excitement factor. Also, the player here is certain that if he fails often he will succeed, which could be a possible exploit.

Good for: Competitive games
Bad for: Drama (single-player games)


CONCLUSION
Unfortunately, there is no right answer. You choose how much randomness is suitable for your game and how it should be biased. The more random it is, the more vulnerable to luck it is. But a completely predictable game, like chess could be boring.

Just remember the two cases when it shouldn't be random:
- Early advantage
- Cumulative advantage

It's all up to you to figure out the right balance for your game.