Maths:
I'm trying to do it correctly right away, but I'm not 100% sure if I got it right. Nonetheless here is what I did.
I compared five worlds:
a) The world where we have 3 Bodyguards and 1 Tracker
b) The world where we have 2 Bodyguards and 1 Tracker
c) The world where we have 1 Bodyguard and 1 Tracker
d) The world where we have 3 Bodyguards and 0 Tracker
e) The world where we have 2 Bodyguards and 0 Tracker
And tried to compute the a priori chance for each of them to be true. Doing this per real maths is way too complex I believe, so I simulated. That means running trial randomizations and tossing a coin into bowl 1 if a) happened to be true; a coin into bowl 2 if 2b happened to be true; etc. Stop at 100 000 coins.
To simplify, I reduced the pool of possible roles to just Bodyguard and Tracker. This shouldn't change anything because every run where another role is chosen would not count towards any of the bowls. I also removed Loud and Witness from the list of modifiers, because we seem to have none of those. Claim if this isn't true.
Results:
World a) has 24% of being true
World b) has 50% of being true
World c) has 21% of being true
World d) has 5% of being true
World e) is actually impossible (duh)
This means, purely based on probability, of the Bodyguards
Two are lying: ~1/4
One is lying: ~1/2
None is lying: ~1/4