There are two marbles we can label PUT_IN and MYSTERY_MARBLE, but since MYSTERY_MARBLE may be either black or white, we can call these alternatives MYSTERY_MARBLEb and MYSTERY_MARBLEw. (Since PUT_IN is known to be white, it can simply be PUT_IN.) There are four possible outcomes of our first drawing:
1) Marble PUT_IN is drawn, leaving MYSTERY_MARBLEw in the bag;
2) Marble PUT_IN is drawn, leaving MYSTERY_MARBLEb in the bag;
3) Marble MYSTERY_MARBLEw is drawn, leaving PUT_IN in the bag;
4) Marble MYSTERY_MARBLEb is drawn, leaving PUT_IN in the bag;
But since a white marble is drawn, (4) did not occur. So there are three possible outcomes, two of which have a white marble left, so the odds are 2/3.
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