How we determine facts.

[h4ym4n]

Consider the following.

Suppose you have a bag of marbles. The bag may contain only marbles colored white or it may not. You do not know. What is the probability that the bag contains only white marbles. You don't know, so lets say the probability the bag only contains white marbles is X.

Suppose you draw a marble from the bag and it is white. The result of this draw doesn't tell you the probability of X, but you do learn something from it. You learn the probability of X is closer to 1 than you previously thought. Why? Because each time you draw a marble and find it to be white, you decrease the number of ways a non white marble could be in the bag.

For example, suppose the bag has 3 marbles in it. W stands for a white marble. B stands for a non-white(black) marble. If 3 marbles are in the bag the possible starting conditions are:
WWW (3 white marbles)
WWB (2 white marbles and black marble)
WBB (1 white marbles and 2 black marbles)
BBB (3 black marbles)

Before we start drawing marbles the probability that all the marbles are white is .25. Now suppose we draw a white marble. Now only two marbles are left in the bag and we've eliminated one possible initial starting condition. The remaining possible starting conditions are:
WWW
WWB
WBB.

We now know that the probability all the marbles are white is .33. Suppose we draw another white marble. The remaining possible starting conditions are:
WWW
WWB

We now know that the probability all the marbles are white is .5. Do you see what is going on here? Every time you observe a white marble while never observing a non-white marble confidence in the proposition that all the marbles are white increases.

It is essentially this thinking that we use to determine what is a fact or at least likely to be true of reality and what isn't.

Isn't this a classic Monty Hall problem? I'm not smart enough to figure that out.

Change the white and black marbles with a goat and a car.