Ken's Puzzle of the Week

More Balls and Urns

  1. There are two empty urns in a room. You have 50 white balls and 50 black balls. After you place the balls in the urns, two random balls will be picked, each from a random urn. The first ball will not be returned to the urns after it is removed.  Distribute the balls (all of them) into the urns to maximize the chance of picking:
    1. two white balls.
    2. one white and one black ball (in any order.)
  2. Repeat problem 1 if the balls will both be taken from the same urn.
  3. What is the smallest number of balls (and their distribution) to make the chance of pulling out two white balls, without replacement, each from a random urn, exactly 1/2?  Exactly 1/3?

Source: Original. Similar to puzzle on February 17, 1998.

Solutions were received from Dan Chirica, Mark Rickert, Denis Borris, Joseph DeVincentis, and Philippe Fondanaiche.  For problems 1 and 2, the best arrangement is to put exactly what you want in one urn and the remaining balls in the other.  Four balls and eight balls solve the respective parts of problem 3.  Philippe's summary is complete:
 
Q1
a - To maximize the chance of picking 2 white (W)  balls, the best way is to put 2 W balls in the urn U1 and 48 W balls + 50 B (black) balls in the urn U2.
There are 4 four possibilities to choose the two urns : U1 & U1, U1 & U2, U2 & U1, U2 & U2, each of them having the same probability 1/4.
So the probability P(1a) to pick the two W balls is equal to:
P(1a) = 1/4 * [2/2 * 1/1 + 2/2 * 48/98 + 48/98*2/2 + 48/98 * 47/97 ] = 10537 / 19012 = 0,554228...
It is easy to check that if the urn U1 contained a higher number of W balls or a certain number of B balls, the probability P(1a) should decrease.
 
b- Same reasoning: we put one W ball and one B ball in the urn U1. So the urn U2 contains 49 W balls and 49 B balls.
So the probability to pick one W ball and one B ball (in this order) is equal to:
1/4 * [1/2 * 1/1 + 1/2 * 49/98 +  49/98 * 1/2 + 49/98 * 49/97]
The probability to pick one B ball and one W ball (in this order) is the same.
So the requested probability to pick one W ball and one B ball (in any order) is equal to:
P(1b) = 1/4 * [ 1 + 2 * 49/98 + 2 * 49/98 * 49/97] = 243 / 388 = 0,626288....
 
Q2
We have two possible ways to choose the urns: U1 & U1 ot U2 & U2, the corresponding probabilities being 1/2 and 1/2.
The compositions of the urns  are respectively the same as in Q1.
Therefore:
a - P(2a) = 1/2*[ 2/2 * 1/1 + 48/98 * 47/97 ] = 5881 / 9506 = 0,618661...
b - P(2b) = 1/2 *[1 + 2* 49/98 * 49/97] =73 / 97 = 0,752577....
 
Q3
The smallest number of balls to make chance P of pulling out 2 W balls without replacement each from a random urn exactly 1/2, is equal to 4 balls with a distribution of 2 W balls in the urn U1 and 1 W ball and 1 B ball in the urn U2. 
Indeed P = 1/4 * [ 2/2 * 1/1 + 2/2 * 1/2 + 1/2 * 2/2 + 1/2 * 0] = 1/2
If P = 1/3, we put 2 W balls in the urn U1, 1 W ball and n B balls in the urn U2 such that
1/4 * [2/2 * 1/1 + 2/2 * 1/(n+1) + 1/(n+1) * 2/2 + 1/(n+1) * 0 ] = 1/3
Therefore n = 5. Then there are 8 balls with 3 W balls (2 of then in U1, the 3rd in U2) and 5 B balls all of them in U2.
Mark Rickert provided a table for the total number and distribution of balls for each probability up tp 1/50: 
3.  Below is a table for 1/1, 1/2, 1/3, ..., 1/50.
 
      URN1      URN2
1/N  B    W    B    W    total
------------------------------
 1   0    2    0    2      4
 2   0    2    1    1      4
 3   0    2    5    1      8
 4   0    2    2    0      4
 5   1    3    4    1      9
 6   1    1    2    2      6
 7   1    1    4    3      7
 8   1    1    1    1      8
 9   1    2   11    1     15
10   1    1    4    2      8
11   4    1    6    5     16
12   1    1    2    1      5
13   5    1    8    6     20
14   2    0    3    4      9
15   2    1    7    3     13
16   1    1    3    1      6
17   2    0    9    9     20
18   2    1    2    1      6
19   2    0   10    9     21
20   1    1    4    1      7
21   3    1    5    2     11
22   2    0    6    5     13
23  10    1   15    8     34
24   2    0    2    2      6
25   6    4   29    1     40
26   3    1   10    3     17
27  14    8   26    1     49
28   1    1    6    1      9
29   7    1   21    8     37
30   2    1    4    1      8
31   2    1   26    5     34
32   3    1    3    1      8
33   7    3   10    1     21
34   1    1   16    2     20
35   4    2   13    1     20
36   2    1    5    1      9
37   8    1   28    9     46
38   1    1   18    2     22
39   2    0   18    9     29
40   2    0    3    2      7
41  22    6   73    9    110
42   2    1    6    1     10
43   5    1   35    8     49
44   1    1   10    1     13
45   5    1    8    2     16
46   1    1   22    2     26
47  19    1  106   35    161
48   3    1    5    1     10
49   ?    ?    ?    ?      ?  nothing found for up to 400 of each color in each urn
50   1    1   24    2     28
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