Dmitry Chernikov’s Blog

Suppose there are 100 balls in the urn. Each ball is either red or black, but you don’t know how many balls are of what color. You start taking the balls out of the urn one by one. Each time you get a red ball. What is the probability after you have taken out 99 balls, each of which has turned out to be red, that the last ball you take out will also be red?

Or, in other words, does that fact that the 99 balls were red somehow increase the chances that the last ball, too, will be red (e.g., because if it were black, chances are, we would have seen it among the first 99 balls)?

Again, suppose that there are three balls in the urn. The hypothesis is that one of them is black. The evidence is that two balls we have taken out so far turned out to be red. To what extent does the evidence support the hypothesis? The probability of getting the black ball at first is 1/3. The probability of getting it the second time is 2/3*1/2 = 1/3. The total probability is 2/3. So, if the last ball is black, the probability of getting two red balls in a row is 1/3. Similarly, if the last ball is black, then getting 99 red balls in a row would be extremely unlikely (1%) and count against the hypothesis that the 100th ball is black.

This is Carnap’s “inductive probability.”