# Planet Brilliantia

Probability Level 3

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

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

It is also known that there is a $\frac{2}{3}$ chance a creature from Brilliantia is a mathematician and a $\frac{1}{3}$ 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 $\frac{a}{b}$, for some coprime positive integers $a, b$, find the value of $b - a$.

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