That can't be right. The possibility of successfully navigating an asteroid field is approximately 3,720 to 1.
The ecliptic plane, I believe, is defined as the plane of earth's orbit, i.e. we orbit at 0 degrees inclination. I'm pretty sure we're the only planet at exactly 0 degrees. Mercury is like 7 degrees, and pluto is way out.
I'm pretty sure we're the only one at 0 degrees because it's arbitrarily set that way. I mean it only makes sense to do so, unless the ecliptic plane is exactly perpendicular to the rotational axis of the Sun, which I have no idea whether or not it is. If it is, then go us!
Although the solar system resides within one plane for the most part, going up or down in the Z direction, you would have an equal chance of hitting a planet as in any other direction, because the universe is actually very uniformly distributed in all directions. The cosmic microwave background radiation tells us so.