Here is that “fancy” trick I promised to teach you. All that you do is multiply the number of outs you have on the flop by 4 with two cards to come and by 2 with one card to come. Voilà, you have your answer. Now, let’s test it.
Let’s say you have a gut shot straight draw on the flop which gives you 4 outs. 4 times 4 equals 16, so you can expect to hit your gut shot 16% of the time by the river. 4 times 2 is 8, so you can expect to hit your gut shot 8% of the time by the turn. The actual numbers are 16.5% and 8.7%, but the results are close enough to make intelligent decisions.
Your chance of improving also comes pretty close to telling you what your likely equity is if you were to get all-in. Just remember to keep in mind that the calculation only works if you actually have your expected number of outs. For example, if you shove all-in with two overs expecting to have 24% equity and get called by two-pair or better, the calculation simply does not work. You will find, however, that draws almost always have the number of outs you anticipate when all-in.
This is why professional players tend to play draws so aggressively; they know that no matter what happens, they have a reliable number of outs. When combined with fold equity, aggressive play with draws can be very profitable.