Yes it does, but that really works best if you have a higher spec on your horse.
Horse A has entered a total of 2000 shows and has a spec of 20% (that is 400 out of 2000 shows) for Racing type.
Horse B has entered a total of 500 shows and has a spec of 60% (that is 300 out of 500 shows) for Racing type.
When both horses enter a Racing show, they get some imaginary points. Whoever has more, will be ranked higher on that show:
Pha (points horse A) = (5/100)*2000 + (20/100)*400 = 100 + 80 = 180 points
Phb (points horse B) = (5/100)*500 + (60/100)*300 = 25 + 180 = 205 points
So quite surprisingly, Horse B has better chances to win the show, although it hasn't entered as many shows. That's happening because Horse B's spec is higher.
If you did not get the above formulas, remember this. The better your spec is, the higher the chance you get to win in a show!