Conditional probability problem - Authors writing books I'm currently trying to solve a conditional probability problem and I'm not sure how to tackle it. > Probability that an author published a book about the stock market is $10\%$. > > Probability an author understands topic given the information that he published a book about the stock market is $1\%$. > > Probability that an author understands topic, wrote a book about the stock market and believes he can make better stock market decisions than his dog is $0.001\%$. > > What is the probability that one believes to make better decisions than his dog assuming he understands the topic and wrote a book about the stock market? So, we have: * $A =$ Author wrote a book about the stock market * $B =$ Author understands the stock market * $C =$ Author believes he makes better decisions than his dog We know $\mathbb{P}(A) = 0.1$ and $\mathbb{P}(B|A) = 0.01$. But how do I proceed from here?

We also know that $\mathbb P(A\cap B\cap C)=0.001\%=10^{-5}$. What we need to calculate is $\mathbb P(C|B\cap A)$. \begin{align*} \mathbb P(C|B\cap A)&=\frac{\mathbb P(A\cap B\cap C)}{\mathbb P(B\cap A)}\\\&=\frac{\mathbb P(A\cap B\cap C)}{\mathbb P(A)}\cdot\frac{\mathbb P(A)}{\mathbb P(B\cap A)}\\\&=\frac{\mathbb P(A\cap B\cap C)}{\mathbb P(A)}\cdot\frac{1}{\mathbb P(B|A)}\\\&=\frac{10^{-5}}{10^{-1}}\cdot10^2\\\&=10^{-2}=1\%. \end{align*}