My favourite example of motivated stopping is Lazzarini’s experimental “verification” of the Buffon needle formula.
(Drop toothpicks at random on a plane ruled with evenly spaced parallel lines. The average number of line-crossings per toothpick is related to pi. Lazzarini did the experiment and got pi to 6 decimal places. It seems clear that he did this by doing trials in batches whose size made it likely that he’d get an estimate equivalent to pi = 355⁄113, which happens to be very close, and then did one batch at a time until he happened to hit it on the nose.
Completely off-topic, here’s a beautiful derivation of the formula: Expectations are additive, so the expected number of line-crossings is proportional to the length of the toothpick and doesn’t depend on what shape it actually is. So consider a circular “toothpick” whose diameter equals the spacing between the lines. No matter how you drop this, you get 2 crossings. Therefore the constant of proportionality is 2/pi. Therefore the expected number of crossings for any toothpick of length L, in units where the line-spacing is 1, is 2L/pi. If L<1 then this is also the probability of getting a crossing at all, since you can’t get more than one.)
My favourite example of motivated stopping is Lazzarini’s experimental “verification” of the Buffon needle formula.
(Drop toothpicks at random on a plane ruled with evenly spaced parallel lines. The average number of line-crossings per toothpick is related to pi. Lazzarini did the experiment and got pi to 6 decimal places. It seems clear that he did this by doing trials in batches whose size made it likely that he’d get an estimate equivalent to pi = 355⁄113, which happens to be very close, and then did one batch at a time until he happened to hit it on the nose.
Completely off-topic, here’s a beautiful derivation of the formula: Expectations are additive, so the expected number of line-crossings is proportional to the length of the toothpick and doesn’t depend on what shape it actually is. So consider a circular “toothpick” whose diameter equals the spacing between the lines. No matter how you drop this, you get 2 crossings. Therefore the constant of proportionality is 2/pi. Therefore the expected number of crossings for any toothpick of length L, in units where the line-spacing is 1, is 2L/pi. If L<1 then this is also the probability of getting a crossing at all, since you can’t get more than one.)