I found the following example of a case where Bayes' theorem could be used, on this French wikipedia page. No solution has been given though.
Contradictory Forecasts
- According to weather station $A$, the weather will be nice tomorrow.
- Another station $B$ predicts rainy weather.
- It is known that A makes errors in $25\%$ of all cases, and B in $30\%$ of all cases.
- It is known that on average, $60\%$ of all days are rainy and $40\%$ are nice.
Who is right with what probability?
I think I would be able to give one if you see what version of the theorem should be applied, so a small hint would be enough to help me to give an answer. However, feel free to give a complete answer if you like.