The Bayesian Reasoning Helper

Help me update my belief about this given new evidence.

What I'm trying to judge: [DESCRIBE — E.G. IS THIS LEAD REAL, IS THIS
TEST RESULT MEANINGFUL]
My prior sense of the odds: [DESCRIBE]
The new evidence: [DESCRIBE, WITH HOW RELIABLE IT IS]

Provide:
1. How to combine my prior with the new evidence properly
2. Why the base rate matters as much as the new signal (the classic error)
3. A rough updated probability, with the reasoning
4. How a rare underlying event makes even a good test produce false
   positives
5. What it means for the decision

Show me why ignoring the base rate leads to overconfident conclusions from
a single positive signal.