Given the random orientation of planetary orbits relative to our line of sight, do you expect most or only a minority of all planetary systems to show transits?

To determine whether most or only a minority of all planetary systems show transits, we need to understand a few concepts: planetary orbits, line of sight, and transits.

Planetary orbits: Planets orbit their host star in a path called an orbit. These orbits can have various orientations relative to our line of sight.

Line of sight: The line of sight refers to the direction from which we observe a planetary system. It is the line connecting us to the system.

Transits: A transit occurs when a planet passes directly between its host star and the observer (us in this case). It results in a small decrease in the star's brightness when the planet blocks a portion of the star's light.

Now, back to your question. Since the orientations of planetary orbits are random relative to our line of sight, the chance of a random planetary system showing transits depends on the alignment between the planetary orbit and our line of sight.

If we consider a large number of planetary systems across the universe, the orientations of their orbits will be distributed randomly. Consequently, some planetary systems will have orbits aligned in such a way that they produce transits visible to us, while others will not.

Therefore, based on the random distribution of planetary orbits, we can expect that only a minority of all planetary systems will show transits, while the majority will not. The exact fraction will depend on the statistical distribution of orbital orientations.

Note that this answer is based on statistical probabilities and assumptions about the distribution of planetary systems in the universe. As our knowledge of exoplanets evolves, more precise estimates may become available.