The problem is like that: "John wakes up with a sore throat. Use Bayes’ rule to infer the posterior probability distribution for what John’s ailment is. Show your final result to three decimal places"
I am not given $P(\text{sore throat})$, does it mean that as it is true, its probability is 1?
My solution:
$$P(\text{hangover} \mid \text{sore throat}) = \frac{P(\text{sore throat}\mid \text{hangover}) \cdot P (\text{hangover})}{P(\text{sore throat})} = \frac{0.1\cdot 0.3}{????}$$ Note: $0.1$ and $0.3$ are given
Hint apply total probability: $$P(\text{sore throat})=P(\text{sore throat}\mid \text{hangover}) \cdot P (\text{hangover})+P(\text{sore throat}\mid \text{no hangover}) \cdot P (\text{no hangover})$$ where $P (\text{no hangover})=1- P (\text{hangover})=0.7$. $P(\text{sore throat}\mid \text{no hangover})$ should be given.