And counting bet twice only in case of horse winning is equivalent of betting with 2:1 odds. Bookmaker will only give such odds if probability of that horse winning is 1⁄3. Hence the 1⁄3 probability.
You can model it as having 2:1 odds or 1:1 odds with the bet counted twice. The later requires a trivial change to your betting algorithm. It also has the advantage of not changing your probability of a horse winning due to a mistake in a bookmaking system.
And counting bet twice only in case of horse winning is equivalent of betting with 2:1 odds. Bookmaker will only give such odds if probability of that horse winning is 1⁄3. Hence the 1⁄3 probability.
You can model it as having 2:1 odds or 1:1 odds with the bet counted twice. The later requires a trivial change to your betting algorithm. It also has the advantage of not changing your probability of a horse winning due to a mistake in a bookmaking system.
But that is not an actual analogy to sleeping beauty. Real analogy would be, that you are a “counted bet”, what horse are you more likely to be on?