Probability
# Conditional Probability

One green ball, one blue ball, and two red balls are placed in a bowl. I draw two balls simultaneously from the bowl and announce that at least one of them is red.

What is the chance that the other ball I have drawn out is also red?

Box I contains 1 red ball and 2 white balls. Box II contains 2 red balls and 1 white ball. One ball is drawn randomly from box I and transferred to box II.

Then, a ball is drawn randomly from box II and it is red. What is the probability that the transferred ball was white?

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 $b - a$.