In which case, for every planet that we detect, there should be significantly more that we can't detect because they don't transit in line with the Earth and, thus, are invisible. Now, I suppose that we can reasonably assume that most planets in a given solar system orbit in a fairly well defined plane, so if we detect one planet then there are probably others in the same plane. But, if we have detected no planets in a system, isn't it simply more likely that the planets are orbiting in a plane that doesn't bisect the Earth? If that is the case, since there is such a tiny degree of variation that would allow a transit to be observed, shouldn't there be a huge number of planets that are not visible because the inclination of their orbital plane is something other than 0 (with respect to the Earth)?
Isn't it likely that most extra solar planets don't transit their star in line with the Earth?
Yes this is certainly true. The chances that a planet's orbital plane is directly along our line of sight is pretty small. And you are also correct to assert that it is more likely that most planets will orbit in the same plane. So if we don't see a planet around a star, it doesn't mean the planets don't exist. Absence of evidence is not evidence of absence.
In the early days of exoplanet detection, we had only a handful of known planets to study. This hardly precludes itself to statistical analysis. However, as the years have progressed, the number of known and confirmed planets has piled up, allowing astronomers to start inferring information based on statistical arguments.
One can argue that if we've spotted X number of planets after observing Y number of stars under Z conditions for T time then with a little bit of statistical analysis and math, there should be A number of planets in our galaxy, many of which we can't observe with current techniques. In fact, that is precisely what many astronomers have done.
Cassan et al. (2012) looked at microlensing detections and were able to infer, based on known detections and statistical analysis, that:
- Planets are at least as numerous as the stars in the Milky Way.
- 17% of stars host a Jupiter-mass planets ($0.3$-$10\:M_J$).
- 52% of stars host "Cool Neptunes" ($10$-$30\:M_\oplus$).
- 62% of stars host an Earth-mass planet ($5$-$10\:M_\oplus$).
Swift et al. (2013) performed a similar type of analysis but focused mostly on M Dwarfs. They inferred that M Dwarf stars are expected to have, on average, $\sim2$ planets per star.
There are certainly many more such analyses and almost all of them come to the conclusion that there are hundreds of billions of planets in our galaxy. This can be confidently stated based on only the ~3500 exoplanets found so far.