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From what I understand, the TRAPPIST 1 system was discovered using the transiting method looking at 20 ultra-cool small stars. I'm not sure how many have been observed since this study.

What I wonder is, what would we expect to see if all 20 had planets? I'm not sure how many planetary systems in this and future similar studies we would expect to see edge on. What about radial velocity measurements?

Did we just get lucky with TRAPPIST 1?

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The probability of observing a planet transit is approximately $(R_p + R_s)/a$, where $R_p$ and $R_s$ are the planet and stellar radius respectively and $a$ is the semi-major axis of the planet's orbit. This assumes that planetary orbits are circular and randomly oriented with respect to our line of sight to the star (for which there is little or no counter-evidence).

In the case of Trappist-1, it has an estimated radius of $R_p=0.11R_{\odot}$, the planetary companions have radii and semi-major axes of (in units of Earth radii and au respectively) (1.09,0.011), (1.06, 0.015), (0.77,0.021), (0.92,0.028), (1.05, 0.037), (1.13, 0.045), (0.75, 0.059).

Using approximation above, then the transit probabilities are 0.051, 0.037, 0.026, 0.020, 0.015, 0.012, 0.009 respectively.

Now these probabilities cannot be treated as independent of one another, since it is quite likely (though this is still the subject of research) that planets will naturally tend to inhabit the same orbital plane (the Trappist-1 planetary orbit inclinations are within 0.2 degrees of each other - flatter than the solar system). Thus you are much more likely to see second and subsequent transits in a system with one transiting planet than you are in a star picked at random. On the other hand, a system with lots of planets that are not quite in the same orbital plane gives an increased chance of seeing one of them transit.

Either way, you can see from the probabilities above that seeing one and only one transiting system of this type from 20 targets, even if all of them had similar planetary systems, is not at all unexpected.

Radial velocity studies of such systems are difficult, but currently being attempted. Trappist-1 has an apparent J magnitude of 11.4, which is very challenging for current telescopes and instrumentation to get high resolution spectroscopy.

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