There are a few true white horses, but they are few and far between. Generally speaking, most white horses are born ga darker colour and get whiter with age. Dessie was bay when he was born but he evidently inherited a grey gene from each of his parents (his sire was grey and his second dam on his mother's side was also grey). To turn grey a horse needs to inherit two copies of this gene.
To go over some basic genetics, every gene inherited has two chromosomes, 1 inherited from each parent. With regards to colour, the Grey gene is a recessive gene which comes in two forms G and g. If a copy of the G gene is present then the horse will always be Grey, if the horse gets 1 copy of the g gene he will not be grey, but if he gets 2 copies of the g gene he will be grey. With me so far?!
If you look at Silver Patriarch as a racehorse, he was much darker:
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Whereas as he got older, at stud, he became much lighter:
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Markings are a little more difficult to explain, it is all to do with the migration of melatonin within the foetus whilst in utero. The migration is also dependent on the position of the foetus, hence why full siblings never carry identical markings.
Does that help?!