Q: An urn contains one hundred white and black balls. You sample one hundred balls with replacement and they are all white. What is the probability that all the balls are white?

A: This question cannot be answered with the information given.

In general, the following formula gives the conditional probability that all the balls are white given you have sampled one hundred balls

- and they are all white
P(100 white | 100 white samples) =

P(100 white samples | 100 white) * P(100 white)

sum(i=0 to 100) P(100 white samples | i white) * P(i white)

The probabilities P(i white) are needed to compute this formula. This does not seem helpful, since one of these (P(100 white)) is just what we are trying to compute. However, the following argument can be made: Before the experiment, all possible numbers of white balls from zero to one hundred are equally likely, so P(i white) = 1/101. Therefore, the

- odds that all 100 balls are white given 100 white samples is
P(100 white | 100 white samples) =

1 / ( sum(i=0 to 100) (i/100)^100 ) =

63.6%

This argument is fallacious, however, since we cannot assume that the urn was prepared so that all possible numbers of white balls from zero to one hundred are equally likely. In general, we need to know the P(i white) in order to calculate the P(100 white | 100 white samples). Without this information, we cannot determine the answer.

This leads to a general "problem": our judgment about the relative likelihood of things is based on past experience. Each experience allows us to adjust our likelihood judgment, based on prior probabilities. This is called Bayesian inference. However, if the prior probabilities are not known, then neither are the derived probabilities. But how are the prior probabilities determined? For example, if we are brains in the vat of a diabolical scientist, all of our prior experiences are illusions, and therefore all of our prior probabilities are wrong.

All of our probability judgments indeed depend upon the assumption that we are not brains in a vat. If this assumption is wrong, all bets are off.