Imagine you are a radar operator aboard a submarine and you have a sensor that alerts you of incoming torpedos. The sensor has a 90% hit rate (it goes off 90% of the times there is a torpedo coming). It has a 10% false alarm rate. You know that you are currently in friendly waters, so your subjective prior probability of being hit at any given moment is low: .05. In this situation, what is the probability of being hit given that the sensor goes off?
I solved this as: $$P(\text{Hit}\mid \text{Alarm}) = \frac {P(\text{Alarm}\mid \text{Hit}) \times P(\text{Hit})} {P(\text{Alarm})}$$ Which becomes: $$P(\text{Hit}\mid\text{Alarm}) = \frac {.9 \times .05} {.1} = .45$$
The issue I see is that when you raise the prior, the posterior can become $>1$ which confuses me. Thank you for your help, I am new to Bayes Theorem and probability generally so I apologize if this problem is too simple.