Probability: 3 balls, 2 Urns I have this question which I can't get my head round. I know what the answer is now, but I don't get how it's calculated. Here's the question:
Three distinct balls are distributed randomly into two distinct urns. What is the probability that one of the urns contains one ball and the other urn contains two balls?
A) 5/6
B) 1/2
C) 3/4 (Answer)
D) 2/3

I'm not all that great with probabilities and always find a hard time grasping the simple of concepts but for the above question, I get:
Urn1   Urn2
3      0
2      1
1      2
0      3

So, I assumed it was 2/4 (which wasn't in the answer option) so simplified it to 1/2 which was wrong. (Not sure if 'simplification' is allowed in probabilities...)
Can someone explain please?
Thanks.
 A: The key word is distinct. Let's say we have a red, blue, and green ball (r,g,b for short). We now have to figure out all the possible combinations in each urn.
   Urn 1      Urn 2
1. -          r, g, b 
2. r          g, b
3. g          r, b
4. b          g, r
5. r, g       b
6. r, b       g
7. b, g       r
8. r, g, g    -

So now, 6 of the 8 possibilities have two balls in one urn and one in the other. Reducing that the probability is 3/4. If the question instead said 3 indistinguishable balls your answer would have been correct.
A: What is the opposite of what's being asked?  You're interested in the balls being split up into the urns in a 2/1 or 1/2 configuration.  The opposite of that is a 0/3 or 3/0 configuration.  In other words, what is the probability that all three balls are in the same urn?  That's easier to calculate.
$P(B_1 Left) = 0.5$
$P(B_2 Left) = 0.5$
$P(B_3 Left) = 0.5$
$P(all Left) = 0.5 * 0.5 * 0.5 = 0.125$

$P(B_1 Right) = 0.5$
$P(B_2 Right) = 0.5$
$P(B_3 Right) = 0.5$
$P(all Right) = 0.5 * 0.5 * 0.5 = 0.125$

Adding together:
$P(all Same) = 0.125 + 0.125 = 0.25$
$P(\overline{all Same}) = 1 - P(all Same) = 1 - 0.25 = 0.75$
