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The equations are $Y = \frac{1}{2} + \frac{1+x}{2}$ and $X = 1+y$

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If the probability of heads is $p_i$ for coin $i$ that has prior probability of $1/3$, then the posterior probability for that being the correct coin assuming you get $n$ heads out of $7$ is proportional to prior times data likelihood, which is $(1/3)*p_i^n*(1-p_i)^{7-n}$. Once you get these 3 posterior probabilities for the 3 coins you can divide them by ...

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If $P(A|B,C)$ means the probability that $A$ happens assuming both $B$ and $C$, it is just $P(A|B\cap C)$ written differently. Now you can just use definition several times: \begin{align} P(A|B\cap C) &=\frac{P(A\cap B\cap C)}{P(B\cap C)}=\\ &=\frac{P(B|A,C)P(A\cap C)}{P(B|C)P(C)}=\\ &=\frac{P(B|A,C)P(A|C)P(C)}{P(B|C)P(C)}=\\ ...

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