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I've never seen any information about the distribution of biomes in a generated world.

First, are all types of biomes, guaranteed to exist in a world? Will I always find snow somewhere, and desert somewhere? I've got one world that doesn't seem to have cactus (so I guess the sand is beaches, no actual desert).

Second, is there any kind of pattern or relation to biomes? Is snow to the North, and desert South, for example?

Third, how about density of biomes? If all types do exist, would I find them all within distance X of my spawnpoint, or have to travel for weeks to find a given type?

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Since the world of Minecraft is infinite, (see Notch's blog post about that interesting subject,) all worlds should have all biomes, if you search far enough for them. It is possible that some seeds randomly generate very large biomes, but this is only due to the random nature of terrain generation.

For versions starting with 1.7 snapshots

From Jeb's blog post:

Biomes have been put into four main categories: snow-covered, cold, medium, and dry/warm. Biomes will avoid getting placed next to a biome that is too different to itself (sometimes this still happens, but it’s very rare now and not all over the place)

[...] Most biomes have uncommon/rare variations that you may run into.

For versions starting with Beta 1.8

In Beta 1.8, biomes got an overhaul. They are no longer determined by randomized metrics such as rainfall and temperature. Rather, they appear to be assigned randomly to fractal sections of the world, as determined by the world's seed.

This was done to better allow new biomes to be inserted. Rather than changing the entire temperature/rainfall simulation, new biomes can simply be generated when new chunks are explored. This is also why biome information is now stored in the anvil file format itself, rather than regenerated every time the game is run. In this way, even if generation code changes occur, currently-explored biomes should not change.

Lastly, new "technical" biomes were introduced to support transitioning from one specific biome to another.

For versions prior to Beta 1.8

Biomes are defined using different aspects of environment such as a rainfall and temperature, which presumably are defined in a similar manner to height (ie. a Perlin Noise map overlaying the world.) These are used to determine the biome for that area, with deserts being hot and dry, rainforests being hot and wet, etc. The Minecraft Wiki has some great illustrations of biomes, with one of the most informative being this one:

Minecraft Biomes

Assuming rainfall and temperature are evenly distributed, the distribution of each biome should be relative to their size in the graphic above. (This may not be a true assumption, though, and would take some digging into the source code to tell.)

To answer your final question, "North" and "South" are abstract and irrelevant, due to the infinite nature of Minecraft. Minecraft worlds do not simulate a planet, but a massive, flat world.

For those of you interested in a more procedural biome-generation method, check out dungeonleague.com, and this post and few preceding posts in particular.

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    Well, by North and South, I really meant, "perpendicular to Sunrise and Sunset" - which are not abstract. In other words, would I be more likely to find a given biome in any given direction, or is it just random? Based on your post, it sounds random. Guess I just got unlucky in my quest for Cacti... :)
    – Cyclops
    Commented Jul 21, 2011 at 13:59
  • Still random, as far as I can tell. Sunrise and Sunset are not abstract, true, but just as irrelevant. I've not seen any indication the direction (as opposed to angle) of the sun has any effect on the game play. I knew what you meant. =]
    – dlras2
    Commented Jul 21, 2011 at 14:04
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    @Cyclops — North and South really are abstract, to the point where the players are wrong about where they are! Notch has admitted that, in the Minecraft universe, the sun rises in the North and sets in the South — this can be verified with a map, which will show the player facing up (North) when facing the sunrise. ;-)
    – Ben Blank
    Commented Jul 21, 2011 at 14:25
  • FWIW - Perlin noise can't be used to generate biomes directly (well, it could, but they would be terrible). The reason is that biomes are distinct blobs. There can (and frequently are ) intersections between three distinct biomes. Perlin noise doesn't allow this: the values in PN are simply the smoothed interpolations between a matrix of values. If biomes are edges in a graph where the biome seams are nodes, PN would mean that each node would have a maximum of two edges.
    – mattbasta
    Commented Oct 21, 2011 at 0:02
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    Specifically, there are two perlin noise [or similar] functions in play, and any given biome is defined by thresholds in both. When a 'borderline' in one intersects a borderline in the other, you can get a three or four way border between biomes. That is an old graphic, though... 1.8 has mountain and ocean "biomes" [these are somehow distinct from the height map], and 1.9 has giant mushroom biomes.
    – Random832
    Commented Oct 21, 2011 at 14:42
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Minecraft Worlds are generally infinite unless you have one of the first versions of the game. As this is incredibly unlikely, let's assume you don't.

All biomes are guaranteed to spawn in a world if you walk far enough. For example, Mushroom island is an extremely rare biome, but if you walk for a long time (possibly a VERY long time) it will spawn in a world, even though it might take a while. (Usually, these rare biomes can be found within 8K-10K blocks from spawn)

That being said, it may take you a long time to find the desired biome. For example, the Plains Biome is the most common biome in Minecraft, while the Mesa Plateau is one of the rarest. So, I decided to use my own game of Minecraft and just keep walking. I spawned in a birch forest, and with three days of Minecraft walking, I found a Mesa biome among many others.

The spawning of biomes is generated randomly (as is the size of the biome), but if you keep walking you will find any/all biome(s).

As the accepted answer points out, Minecraft Worlds have no form of North and South and just is flat and goes on forever.

The real takeaway from this answer is if you keep walking (or flying) you'll find any biome.

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