Planet Brilliantia

Probability Level 3

On planet Brilliantia, there are two types of creatures: mathematicians and non-mathematicians.

Mathematicians tell the truth 67\frac{6}{7} of the time and lie only 17\frac{1}{7} of the time, while non-mathematicians tell the truth 15\frac{1}{5} of the time and lie 45\frac{4}{5} of the time.

It is also known that there is a 23\frac{2}{3} chance a creature from Brilliantia is a mathematician and a 13\frac{1}{3} chance that it is a non-mathematician, but there is no way of differentiating from these two types.

You are visiting Brilliantia on a research trip. During your stay, you come across a creature who states that it has found a one line proof for Fermat's Last Theorem. Immediately after that, a second creature shows up and states that the first creature's statement was a true one.

If the probability that the first creature's statement was actually true is ab\frac{a}{b}, for some coprime positive integers a,ba, b, find the value of bab - a.

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