How we determine facts.

[JuliaL]

<snip>
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.

Hang on a minute.
I'm going either to be pedantic or dim.....or dimly pedantic.
But you have to specify something about the initial conditions for the marbles, for example, that the probability of any single marble of being white is 0.5 and black is the other 0.5.

Otherwise, in a universe with only white marbles possible starting conditions are:
WWW (3 white marbles) probability 1.0
WWB (2 white marbles and black marble) probability 0.0
WBB (1 white marbles and 2 black marbles) probability 0.0
BBB (3 black marbles) probability 0.0
and the probability of drawing a white marble on your first draw is 1.0.

and you've not gained any information with which to infer what the other possibilities are.

I had problems with the car and the goat too. Until I reasoned it out thusly.
Three doors, one car, two goats:
On the initial pick, you had a 33% chance of picking the car, 66% chance of the goat.
So Monty showed you a goat, then if you had picked a goat (66% chance) and switched you got the car. If you had picked the car (33% chance) and you switched you got the goat. So you switched your 66% chance of getting the goat for a 66% chance of getting the car.
I mnemonicked this with the phrase, "if you won and you switched, you lost and if you lost and you switched, you won."