Hi frens I am currently studying a bit of probability and I have encountered a curious fact that my book couldn't explain.

While doing a couple of exercises I found a little "trick" that could bypass all the shit explained in the definition:

Definition 1.4 If P(E) =p, the odds in favor of the event E occurring are r:s (r to s) where $$r/s = p/(1?p)$$ . If r and s are given, then p can be found by using the equation $$p = r/(r+s)$$ .

The trick is that I can simply get r : s by calculating $$P(E)/P(*E)$$ , where P(*E) is the probability that the event will not occur.

EG: What are the odds that a card chosen at random from a 52-card deck is an ace?
In order to solve it, I assume that out of 52 cards I only have 4 chances to pick up an ace, which means a prob of 4/52 of drawing an ace;
following my "trick", instead of calculating the formula given by the definition, I just impose that the odds are $$p = (4/52)/(48/52)$$ , wherein the latter is the prob that such event will not occur. The result is 1:12, which is the same as in the solution and the one given by the definition's formula.

Can someone help me discover if this is just a case, or confirm that it's actually a property of the odds definition?