Here’s another example of using Bayesian logic without using complicated math formulas. This is an old (but still good) video of a Mormon deconvert. He’s recounting his deconversion story and at one point he brings up a water analogy about how you find “truth”. Here’s what I’ve transcribed:
Somebody could give me this glass of water and tell me that it’s water. But there’s a lot of clear liquids out there and I might actually have a real case that this might not be water. Now most cases when something like a liquid is in a cup it’s water.
A good way to find out if it’s water is to test if it has two hydrogens per oxygen in each molecule in the glass and you can test that. If it evaporates like water, if it tastes like water, freezes like water… the more tests we apply, the more sure we can be that it’s water.
However, if it were some kind of acid and we started to test and we found that the hydrogen count is off, the oxygen count is off, it doesn’t taste like water, it doesn’t behave like water, it doesn’t freeze like water, it just looks like water. If we start to do these tests, the more we will know the true nature of the liquid in this glass. That is how we find truth. We can test it any number of ways; the more we test it, the more we know the truth of what it is that we’re dealing with.
Who knows how many clear liquids there are out there. Quite a bit. Granted, there’s a high prior probability that a random glass of clear liquid sitting on someone’s kitchen table is water, but it could still be a glass of vodka or hydrogen peroxide. Maybe you have no sense of smell so if you tried to freeze it by sticking it in the freezer for an hour and came back and it wasn’t frozen, it might be a safe bet to conclude that it’s not water. But adding any compound to water (like salt or sugar) changes the boiling and freezing point of water (try it out: pour a bit of sugar or salt in a pot of boiling water and watch it stop boiling almost instantly).
There are a number of tests you can apply for any claim, and each test passed increases the probability of veracity. But relying on only one test is to subject yourself to the Prosecutor’s Fallacy; the probability that it’s water given that it freezes in your freezer is not equal to the probability that it freezes in your freezer given that it’s water!