Evidence of absence

"Absence of evidence is not evidence of absence", they say. They're wrong. In this post we'll work through a scenario and show that absence of evidence is in fact evidence of absence. You can see Yudkowsky for more on this topic.

You have a box with 100 bags in it. Each bag has 100 balls in it. 99 of the bags have 100 white balls, the final bag has 99 white balls and 1 black ball.

(Assume all of the bags are indistinguishable from the outside, and all of the balls are the same size and weight etc., and the black ball is at a random position within its bag).

What is the probability that a bag, selected uniformly at random, contains the black ball? You'll give your first probability estimate before the bag is opened, and then balls will be removed from the bag one by one and you can revise your estimate after seeing each ball. What is your strategy?

Your lab assistant rummages around the box and selects a bag uniformly at random. You should agree that this bag contains the black ball with 1% probability, because you know that 1 in 100 bags contain the black ball.

Your lab assistant takes out the first 99 balls. You observe that they're all white. This observation is merely an absence of evidence of the black ball, and, so they say, it is therefore not evidence of absence of the black ball, and your probability estimate should be unchanged, at a 1% chance that the bag contains the black ball.

But while the 99 bags that only contain white balls would provide this observation every time, the 1 bag that contains the black ball would provide this observation only 1 time out of 100.

So now you ought to agree that the probability that the bag contains the black ball is now only about 0.01%, not 1%, and we see that the ongoing "absence of evidence" of the black ball was in fact evidence of its absence after all.

Your lab assistant pulls out the final ball. It's white.