Does the probability that a single-planet system shows transits increase, decrease

or stay the same if the planet has a rather small or a rather large semi-major
axis?

The probability that a single-planet system shows transits depends on the size of the planet's semi-major axis.

To explain why, let's first understand what a transit is. A transit occurs when a planet passes in front of its host star from our line of sight, causing a slight decrease in the star's brightness. This drop in brightness can be detected and used to infer the presence of the planet.

The probability of a transit occurring is primarily determined by two factors: the size of the planet and the size of its orbit.

1. Size of the planet: Larger planets have a higher chance of causing transits because they block more of the star's light when passing in front of it. Therefore, the probability of a transit increases with the size of the planet.

2. Size of the semi-major axis: The semi-major axis of a planet's orbit determines its average distance from its host star. If the semi-major axis is small, the planet's orbit becomes closer to the star, increasing the probability of a transit. Conversely, if the semi-major axis is large, the planet's orbit is farther from the star, reducing the probability of a transit.

So, to directly answer your question:
- If the planet has a rather small semi-major axis, the probability of a transit increases because of the planet's closer proximity to its star.
- Conversely, if the planet has a rather large semi-major axis, the probability of a transit decreases because the planet's more distant orbit makes it less likely to pass in front of the star from our line of sight.

In summary, the probability that a single-planet system shows transits increases with a rather small semi-major axis and decreases with a rather large semi-major axis.