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# Conditional Probability

Conditional probability is the art of updating probabilities based on given information. What is the probability that the sidewalk is wet? And what if we know that it rained a few hours ago?

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 \(b - a\).

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