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*}