Regarding the genetics of black and whites, the black base color (the "base color" being the color other than white) is based on a combination of two separate genes. The one is common with dominant characteristics; the other recessive and very uncommon.
To get the base color, the foal must receive the dominant gene from at least one parent and the seldom-found recessive gene from both parents.
How did the Pintabian Breed come to be?
It is common misconception that the "Pintabian" (pin-TAY-bee-an) horse is simply a cross between an Arabian and a spotted horse of Pinto or Paint breeding. This, however, is not the case.
Because of its dominant characteristics, experts in the field of genetics have known for years that it is possible to breed the popular tobiano spotting pattern into the Arabian breed to produce a "relatively pure" strain of tobiano horses with over 99% Arabian blood (also known as the Pintabian!). As you may already know, the purebred Arabian horse does not come in the tobiano pattern. Also, because of the genetic make-up of the tobiano gene, at least one parent must be a tobiano in order to produce a horse with tobiano markings. The Pintabian breed was developed by continually crossing tobianos back to purebred Arabians, and in order to be eligible for registration in the Pintabian Horse Registry, horses must be over 99% Arabian in blood.
Here's how it works:
1/2 Arabian = 50% Arabian in blood
3/4 Arabian = 75% Arabian in blood
7/8 Arabian = 87.5% Arabian in blood
15/16 Arabian = 93.75% Arabian in blood
31/32 Arabian = 96.875% Arabian in blood
63/64 Arabian = 98.4375% Arabian in blood
127/128 Arabian = 99.21875% Arabian in blood (Pintabian!)
(and so on)
But what does the fraction 127/128 really mean?
A horse that is 1/2 Arabian usually has one parent that is Arabian and one that is not (1 out of 2 parents are purebred Arabian). A 3/4 Arabian usually has three purebred Arabian grandparents and one that is not (3 out of 4 horses in the second generation are purebred Arabian).
Similarly, of the 128 ancestors in the seventh generation of a Pintabian's pedigree, 127 were purebred Arabian. Only one was not. The horse that was not a purebred Arabian, by the way, is the one that contributed the tobiano pattern.
It is because of this high degree of Arabian blood that the Arabian "type" of the Pintabian horse breeds true.