What percent of star systems have orbits in the right orientation for scientists to find exoplanets through the transit method?

The probability of having an exoplanet’s orbit oriented along our line of sight so that there is a transit depends on two things: the size of the star and the size (or radius) of the orbit. The first part is straightforward: The bigger the star, the more likely we’d witness a transit. The second requires some geometric thinking: If the planet is extremely close to the star, then you can imagine that there will be a transit at almost any viewing orientation. On the other hand, if the planet is far away, the probability of having the correct alignment for a transit is low. More mathematically, if we let D be the diameter of the star and a be the distance of the planet from the star (called the semi-major axis), the probability for a transit is D/(2a). This assumes circular orbits and a small planet.

So, there is no single percentage to answer the question because it depends on how big the orbit is. For a planet whose orbital period is just a few days, the probability could be 10 to 20 percent or more. But for an Earth-like world at 1 astronomical unit (our planet’s distance from our star) with a period of 365 days, the probability is only 0.47 percent. Thus, the percentage ranges, roughly, from 0.5 percent to 15 percent for a star like the Sun. Because NASA’s Kepler mission team is most interested in planets within their stars’ habitable zones — where liquid water might exist on these worlds’ surfaces — the probability is close to that half-percent mark.
 
Another important aspect to keep in mind: Transits of planets in large orbits are rare events. You get just one transit per year of a planet in an Earth-like orbit. So, not only is the probability low of ever seeing a transit, but for the vast majority of the time the planet will not be transiting. You have to observe for many years to know for sure.