How rare are the "rare biomes" in Minecraft?

There are differences in the chances. Most of the time, you do not get mesa or jungle biomes because of the fact that there are 61 different biomes. (Source here) Sometimes, it could be due to conditions needed for it the biome to spawn. For example, the mushroom biome can only spawn in an ocean biome and a river is usually seen next to a beach biome. This means that there is around a 2/61 chance for the biome you want to spawn in. If you think you get many plains biomes, look at the F3 menu again. There are 3 different types of Plains biomes(Plains, Plains M and Sunflower Plains), and other biomes also have different variants. Currently, there are 3 types of Mesa Biomes that can spawn in. As for the jungle, there are 5 different types. This also means that you only have a 8% or less chance of a jungle. Also, from the mojang website, they released an image: Notice that some colors are appearing more often that others. This could also mean that there are different chances of biomes spawning. For example, orange, white, green and blue take up majority of the space in the map, while red and purple take up very little space. If the conditions are followed, the blue probably means ocean, green for plains and white for snowy. Red could be mesa, purple could be mushroom islands.

In the minecraft 1.7.2 update, the minecraft wiki also says:

Jungle biomes have been made much rarer.

This could be another reason why you are not getting as many jungle biomes as the other biomes. This is true for the mesa biome as well.

As of now, I cannot read the code for minecraft(Still in SRG names). However, when I can, I will edit the post. Quote from MC wiki:

Biomes are pseudo-randomly generated using the map seed.

It could also be your seed, try going online to search for mesa or jungle seeds.

The map in @UniversalElectricity's answer is from http://chunkbase.com/apps/biome-finder You can also enter seeds and find biome locations at http://mineatlas.com/ (note that for MCPE, these only work with positive seed numbers below 2 billion something.) Biome rarity depends on what it is next to. For example, Savannah spawns next to desert. Deserts can get very large, as can "temperate" biome collections. Some biomes are larger than others - ice plains can get large. Savannah, jungle, and Mushroom island are indeed very rare, but all biomes can usually be found within 8K - 10K blocks of spawn, most often closer. Some biomes are very common, even though they are small - forests, extreme hills, and plains are examples.