On 11/5/2011 10:36 PM, Bob Engelhardt wrote:
Given: an opaque jar with a large number of white & black marbles, same
number of each. If I pick 2 marbles randomly, it seems that the
probabilities depend upon how I do the drawing.

If I pick one marble, then pick another, the probability of drawing one
black and one white marble is 50%. (4 possibilities, 2 of which give 1
white & 1 black.)

That is incorrect.

Suppose there are 100 marbles in the jar, 50 black and 50 white. Your
probability of drawing a white marble first is 50/100 = 0.50, and the
probability that the second marble will be black is 50/99 = 0.505. The
combined probability of these events is 0.50 * 0.505 = 0.2525. It is
equally likely (P = 0.2525) that you will draw a black marble, followed
by a white one. The probability that one or the other of these will
happen is 0.2525 + 0.2525 = 0.5050, or a little more than half, that you
will draw one marble of each color.

If you pick one marble, *put it back*, then pick another, your
probability of picking one of each color is in fact exactly 50%.

If I reach in and pick 2 marbles AT ONCE, the probability is 33%. (3
possibilities, 1 of which gives 1 of each color.)

That also is incorrect. Just because there are only three possibilities
does not mean they are equally likely. In fact, they are not: there is
only one way you can draw two white marbles, and only one way you can
draw two black ones, but there are *two* ways you can draw one of each
color: white-black, and black-white. It is exactly twice as likely that
you will have one of each color, than that you will have two white ones.

It just doesn't SEEM right that the probabilities could be different.

It doesn't seem right because it isn't right.

Why is it different? Or is it not different?

It is different, but not in the way you suppose (as I have attempted to
explain).

Is there really 4
possibilities when drawing 2-at-once?

Yes.

Drawing 3 marbles gives even worse results. One-at-a-time gives:
1/8 probability of all black,
1/8 all white,
3/8 2 black + 1 white, &
3/8 2 white + 1 black.

3 at once gives:
1/4 all black,
1/4 all white,
1/4 2 black + 1 white, &
1/4 2 white + 1 black.
The probability of drawing all same color 3-at-once is twice that of
one-at-a-time!

No, it isn't, actually (although I can easily understand how it seems
like it). Probability is full of pitfalls for the unwary. You found a
few of them. :-)

Is there a statistician in the house?

I'm a math professor. Will that do?