Section 3: How the game provides IV's to an egg

The title is self-explanatory!

SECTION THREE
How the game provides IVs to an egg

Before we start our breeding guide proper, you need to be informed of how exactly the game assigns the IVs to a baby. This actually differs from game to game.
IV Breeding in EDP

The IVs for the baby are first created at random. Then, a random IV from either parent is inherited to the baby. Then, any random IV except an HP IV is selected from either parent and is passed to the baby. Finally, any random IV except an HP or Defense IV is selected from either parent and passed to the baby. These three random IVs may overlap, that is, the same random IV can be chosen to be inherited in the second, or third, IV transfer.

Let's give an example. Suppose Parent A and Parent B have the following IVs. Here, we also display the baby's IVs next to those of its parents:
Parent A IVs Parent B IVs Baby IVs
HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe
21 7 26 15 4 10 5 3 16 30 28 20 - - - - - -

First, the baby's IVs are generated at random:
Parent A IVs Parent B IVs Baby IVs
HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe
21 7 26 15 4 10 5 3 16 30 28 20 18 0 9 14 11 27

Then a parent's random IV is chosen, and passed to the baby. Suppose Parent A's Speed IV is chosen first. So now the baby's IVs are:
Parent A IVs Parent B IVs Baby IVs
HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe
21 7 26 15 4 10 5 3 16 30 28 20 18 0 9 14 11 10

Then a parent's random IV (other than an HP IV) is chosen, and passed to the baby. Suppose Parent A's Defense IV is chosen:
Parent A IVs Parent B IVs Baby IVs
HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe
21 7 26 15 4 10 5 3 16 30 28 20 18 0 26 14 11 10

Finally, a parent's random IV (other than an HP or a Defense IV) is chosen, and inherited to the baby. Suppose Parent B's Special Attack IV is chosen. Thus, the baby IVs finally become:
Parent A IVs Parent B IVs Baby IVs
HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe
21 7 26 15 4 10 5 3 16 30 28 20 18 0 26 30 11 10

Note that the amount of IVs passed from the parents will not always appear to be 3, for the following two reasons. Firstly, the random baby IVs created at the start might match some of the IVs of the parents, which would seem as if the baby got passed more than 3 IVs. Secondly, two or three of the IVs passed to the baby might be in the same stat, which would seem as if the baby got passed less than 3 IVs.

This means that, while it may appear that the baby will inherit between 1 and 6 IVs from the parents, in actuality both parents will always pass a total of 3 IVs. The chances are as follows:
Number of IVs inherited Percentage chance of being inherited
1 2.44%
2 34.41%
3 53.01%
4 9.55%
5 0.58%
6 0.01%

The majority of eggs (about 8 out of 9 eggs) will have 2 or 3 IVs transferred from the parents.

HP and Defense IVs are harder to inherit from the parents than the other IVs. This is different from the other games (see IV Breeding in RSFRLG), where each IV has the same percentage of being transferred to the baby (16.67%). The percentages for Emerald (and DP) are as follows (considering also the fact that the random IVs generated for the baby can end up to be the same as that of one of the parent's IVs):
Stat Probability of that Stat's IV to be inherited
HP 8.02%
Atk 19.55%
Def 13.78%
SpA 19.55%
SpD 19.55%
Spe 19.55%

This means that in EDP, the Attack, Speed, Special Attack and Special Defense IVs have a better probability of being passed to the baby than in RSFRLG, while the HP and Defense IVs have a worse probability of being passed to the baby than in RSFRLG.
2. IV Breeding in RSFRLG

The IVs for the baby are first created at random. Then, three random IVs from either parent are successively inherited to the baby. These three random IVs may not overlap, that is, the same random IV cannot be chosen to be inherited in the second, or third, IV transfer.

Let's give an example to make things clearer. Suppose the parents have the same IVs as before. Let's recapitulate:
Parent A IVs Parent B IVs Baby IVs
HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe
21 7 26 15 4 10 5 3 16 30 28 20 - - - - - -

First, the baby's IVs are generated at random:
Parent A IVs Parent B IVs Baby IVs
HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe
21 7 26 15 4 10 5 3 16 30 28 20 18 0 9 14 11 27

Then a parent's random IV is chosen, and passed to the baby. This is repeated three times.

Suppose Parent A's Speed IV is chosen first to be inherited:
Parent A IVs Parent B IVs Baby IVs
HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe
21 7 26 15 4 10 5 3 16 30 28 20 18 0 9 14 11 10

Then Parent B's HP IV is chosen to be passed to the baby:
Parent A IVs Parent B IVs Baby IVs
HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe
21 7 26 15 4 10 5 3 16 30 28 20 5 0 9 14 11 10

Finally, Parent B's Defense IV is chosen to be inherited. And the baby's IVs thus become:
Parent A IVs Parent B IVs Baby IVs
HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe HP Atk Def SpA SpD Spe
21 7 26 15 4 10 5 3 16 30 28 20 5 0 16 14 11 10

Note that the amount of IVs passed from the parents is, again, not always 3, but this time, there will always be at least 3 IVs that are passed to the baby. As before, though, the random baby IVs created at the start might match some of the IVs of the parents, which would seem as if the baby got passed more than 3 IVs. The chances for each of them happening are these:
Number of IVs inherited Percentage chance of being inherited
3 82.66%
4 16.26%
5 1.06%
6 0.02%
3. Probabilities for IV Breeding

Now, I think you're asking "What is the probability, given two parents with a few perfect IVs (31) in certain stats, for the baby to obtain those same perfect IVs in those stats?" This question is answered later on, in Part III of this guide. There, a list of all the breeding combinations with their probabilities of inheriting IVs will be given. There is also a handy applet that provides you the probability of breeding a baby with the required IVs, starting from any given parents' IVs. You can then simply compare various methods of getting the desired IVs and save as much time as possible in the breeding process.