Post by firstname.lastname@example.org
You're given an urn, full of red and black marbles, and
i) If you draw a single marble, it will be red or black, 50-50
ii) If you draw a pair of marbles (from the full urn), and
they're matched colors, then draw another pair, the
second pair will match with 50-50 chance. (not
necessarily the same color as the first)
How many marbles in the urn?
Condition (i) tells you that the number r of red marbles and the number b of black marbles is the same: r=b.
So let 2n=r+b be the total number of marbles.
From condition (ii), the number of ways to pick a matched pair among the remaining (r-2)+b balls is the same as the number of ways of picking an unmatched pair.
The number of ways of picking a matching pair is
((r-2) choose 2) + (b choose 2)). The total number of ways of picking a pair is ((n-2) choose 2). So you want
((r-2)choose 2) + (b choose 2)) = ((n-2) choose 2) - ((r-2)choose 2) - (b choose 2).
Using b=r and n=2r, you get
2( [(r-2)(r-3)/2] + (r(r-1)/2)) = (2r-2)(2r-3)/2.
( r^2-5r + 6 + r^2 - r) = (4r^2 - 10r + 6)/2
2r^2 - 6r + 6 = 2r^2 - 5r + 3
r = 3.
So there are 6 marbles, three are red, and three are black.
Indeed, note that if that is the case, the odds of picking a red marble is 3/6 = 1/2, and the odds of picking a black marble are 3/6 = 1/2.
And if you pick two marbles of the same color, say red, then there remain in the urn 1 red marble and 3 black marbles. There are (4 choose 2) = 6 possible draws of a pair; three of these involve a red and a black marble, and three involve two black marbles, so the odds that the second pair will match is 1/2.