Background:
Venus cycle is $584$ days
Mars cycle is $780$ days
Pluto cycle, in this case, is 245.5 YEARS
Question:
What are the odds of all three converging at the same degree point on the same day?
This came from an astronomer friend of mine but essentially he was interested to know what is the probability that all three planets converge at the same degree point. I was thinking this may be somewhat complicated if we take into consideration the rotations and speed of which each planet rotates around the sun. On the other hand, wouldn't it just be
$$\left(\frac{1}{360}\right)^3$$
since all planets rotate in a circular sort of shape around the sun hence the $360$. Any suggestions or thoughts on calculating this would be greatly appreciated.
You speak about convergence "at the same degree point", but do we imply that we are watching from the Earth or from the Sun?
Consider we are "staying" at the Sun. As Pluto has rather big inclination: 17,14°, and two other planets have almost zero inclination, then formally it's possible that such event never occurs.
If we do not consider inclination and if we start looking at random moment of time, then the answer is $\frac1{360^2}$.