On Sat, 05 Nov 2011 22:36:27 -0400, 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.)

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

When you hold those marbles in your hand, you know that there are two
marbles in two positions. The marble in _one_ position can be either
white or black: that's two possibilities. The marble in the _other_
position can also be white or black; those two possibilities times the
first two make four.

(This is easier if you just break down and enumerate the positions: 1, 2,
etc.. I was trying to avoid that for pedagogical reasons).

It just doesn't SEEM right that the probabilities could be different.
Why is it different? Or is it not different? Is there really 4
possibilities when drawing 2-at-once?

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!

Is there a statistician in the house?

The same argument holds no matter how many marbles you draw out, or how:
they are still N distinct marbles, so you still have to treat them as
distinct even if you grab them all at once. Consequently, if the jar has
an infinite number of marbles then there are 2^N, a great many of which
are repeats of other possibilities once you jumble the marbles.

Note that the probabilities only go to exactly 2^N as the number of
marbles in the jar approaches infinity: If there are 10 marbles of each
color in the jar and the first one out is black, then the probability
that the second one will be black is 9/19. So the overall probability of
black/black is 10/20 * 9/19, or 90/380, or 9/38 after you reduce. So is
the probability of white/white. The probability of drawing a black/white
is 10/20 * 10/19, or 10/38, as is the probability of drawing a white/
black.

The more draws compared to the total number of marbles the more skewed
the probabilities get: given a jar with five marbles of each color, the
probability of drawing six white marbles ain't 1:64, it's just plain old
0 (unless you have a felt-tip marker in your back pocket).

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