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.)
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?
Bob