Thomas Bayes's essay was read to the Royal Society in 1763, two years after he died; his friend Richard Price sent it in. The rule in it is how to change your mind by the right amount.

Example. A condition affects 1 person in 100. A test catches it 99% of the time, and false-alarms 5% of the time. You test positive. How worried should you be?

P(+) = 0.99 × 0.01 + 0.05 × 0.99 = 0.0594. So P(condition | +) = 0.0099 / 0.0594, which is about one in six.

Most people guess ninety-something. That gap between what feels true and what's true is most of what makes human judgment interesting, and a good part of why we worry about anyone, person or program, sounding confident.