There's a fascinating video on youtube that has a look at gambling in the 1950's. The horse racing section is below.
More money was bet on horse racing than any other game in 1959. Apparently tipsters were all considered "hucksters with worthless information", totes were "computing machines", takeout was maxed out at 15%, and I am unsure when the saying "you can beat a race, but you can't beat the races" popped up, but it was certainly before 1959. The video starts at the horse racing section, so give it a watch if you'd like.
Take a look at the cool video here.