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Red is recessive (e). A red based horse will be ee. One e from mom. One e from dad. Black (the gene is called Extension) is dominant (E). A horse can be EE or Ee. Either will be black or bay. Both of your horse's parents were Ee. Both passed their e and your horse is ee so he is red. In order for a horse to be bay the horse has to carry dominant agouti which restricts where black shows up on the horse. Agouti does not affect red horses. A black horse has recessive agouti (aa) and a bay horse can be AA or Aa. This is only important when breeding. A bay horse that is EEAA will only ever produce a bay based foal no matter what they are bred to. A black EE aa though can have or sire a bay foal if the horse bred to passes an A but will never have/sire a red foal because it will always pass E. A red horse bred to a red will always produce red but a red bred to a black or bay if the genes are present in what they are bred to can produce red, bay or black. Two bays, two blacks or a bay and black bred together if they both carry e can produce red, bay of black as well. No mystery to it. There are many other genes that effect color but those two produce the base color for all horses. The others are what give you other colors. Example - adding in Cream. A single cream gene dilutes the color of the hair. If paired with red is palomino, bay is buckskin and black is smokey black. If a horse inherits two copies of cream then everything is diluted and hair even more so. You have cremello, perlino and smokey cream.
If you look up Mendel's pea plant experiments you will see the explanation on how dominant and recessive genes work.
If you look up Mendel's pea plant experiments you will see the explanation on how dominant and recessive genes work.