## Description

This article is from the Puzzles FAQ,
by Chris Cole chris@questrel.questrel.com and Matthew Daly
mwdaly@pobox.com with numerous contributions by others.

# 394 probability/apriori.p

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?

probability/apriori.s

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 judgments 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.

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