Show answer There's \(1 : 9\) bad vs\. good widgets\. Bad vs\. good widgets have a \(12 : 4\) relative likelihood to spark\. This simplifies to \(1 : 9\) x \(3 : 1\) = \(3 : 9\) = \(1 : 3\), 1 bad sparking widget for every 3 good sparking widgets\. Which converts to a probability of 1/\(1\+3\) = 1/4 = 25%; that is, 25% of sparking widgets are bad\. Seeing sparks didn't make us "believe the widget is bad"; the probability only went to 25%, which is less than 50/50\. But this doesn't mean we say, "I still believe this widget is good\!" and toss out the evidence and ignore it\. A bad widget is relatively more likely to emit sparks, and therefore seeing this evidence should cause us to think it relatively more likely that the widget is a bad one, even if the probability hasn't yet gone over 50%\. We increase our probability from 10% to 25%\.
I'm failing to grasp how the probability conversion works and so some further explanation may be needed