Are the planets in our solar system all on the same plane?
Not at all, for more than four planets..
Relative to the Sun, the planets are in their orbits, in different planes through the Sun.
At any instant, thee planets will be in a plane and this plane turns about the Sun, changing inclination to ecliptic.
A 4th planet has to cross this plane.
Yet, the probability of the 4th, 5th, 6th, 7th and 8th crossing this plane at the same instant of time is likely to be 0. In my opinion, It is a conjecture that this probability is 0.
It will be an exciting animation to show the motion of a 3-planet triangle and its plane and watch the other planets crossing, it at different times.