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Dear Scott,
This combinatorics problem beats me: There are 60 red balls, 32 blue balls, 19 green balls and 7 yellow balls. In total 118 balls.
Then 97 of the balls are drawn and placed at random into nine different urns. So, the two questions are:
1. What is the probability that each urn will only contain balls of a single color?
2. What is the probability that each urn will only contain balls of a single color apart from one ball that is placed in an urn with balls of another color?
I think your running record is very fast and relevant. A little exercise keeps the mind alive.

This took me awhile, and I'm not sure this can be answered exactly.
However, after some thought, this is what I got:

I did use Excel to formulate the probabilities that 1 urn contained all the balls of the same color.  The chances I got were
Red     0.00132
Blue    4.61E-05
Green   8.74E-06
Yellow  1.28E-06

For nine urns, multiply by 9, giving
Red     0.0119
Blue    0.000415
Green   7.87E-05
Yellow  1.15E-05

Now technically, you have to allow for there being more than one urn with the same color,
but since the chance of one urn having the same color is so small, this is too small too

I have what I did in a spreadsheet that is 126 rows by 13 columns,
but I just cut out the last four columns and 60 rows to send them to you.

Here are the chance of getting all the balls in that urn of the same color
The number of balls is  in  the first column.  The rest of the columns are
the approximate chances for getting Red, Blue, Green, Yellow.
Note that the number at this end have definite space between the columns,
but they most likely won't when you receive them.

1     6.90341E-06     3.68182E-06     2.18608E-06     8.05398E-07
2     2.56684E-05     7.30123E-06     2.57397E-06     3.49375E-07
3     6.30833E-05     9.56998E-06     2.00319E-06     1.00174E-07
4     0.000115274     9.32667E-06     1.15915E-06     2.13560E-08
5     0.00016705     7.20841E-06     5.31934E-07     3.61060E-09
6     0.000199964     4.60198E-06     2.01635E-07     5.04235E-10
7     0.000203353     2.49599E-06     6.49334E-08     5.98245E-11
8     0.000179335     1.17396E-06     1.81335E-08
9     0.000139314     4.86387E-07     4.46082E-09
10     9.65159E-05     1.79716E-07     9.78639E-10
11     6.02295E-05     5.98130E-08     1.93391E-10
12     3.41343E-05     1.80791E-08     3.47072E-11
13     1.76902E-05     4.99708E-09     5.69591E-12
14     8.43283E-06     1.27044E-09     8.59818E-13
15     3.71616E-06     2.98590E-10     1.19986E-13
16     1.52052E-06     6.51585E-11     1.55464E-14
17     5.79858E-07     1.32526E-11     1.87742E-15
18     2.06800E-07     2.52073E-12     2.12028E-16
19     6.91792E-08     4.49730E-13     2.24605E-17
20     2.17651E-08     7.54631E-14
21     6.45574E-09     1.19377E-14
22     1.80915E-09     1.78421E-15
23     4.79951E-10     2.52446E-16
24     1.2075E-10     3.38733E-17
25     2.88573E-11     4.31742E-18
26     6.56062E-12     5.23494E-19
27     1.42085E-12     6.04663E-20
28     2.93502E-13     6.66154E-21
29     5.78941E-14     7.00805E-22
30     1.09165E-14     7.04765E-23
31     1.96962E-15     6.78177E-24
32     3.40354E-16     6.25015E-25
33     5.63761E-17
34     8.95806E-18
35     1.36648E-18
36     2.00244E-19
37     2.82066E-20
38     3.82148E-21
39     4.98237E-22
40     6.25435E-23
41     7.56262E-24
42     8.81238E-25
43     9.8996E-26
44     1.07252E-26
45     1.121E-27
46     1.1307E-28
47     1.10094E-29
48     1.03504E-30
49     9.39809E-32
50     8.24324E-33
51     6.9858E-34
52     5.72093E-35
53     4.52808E-36
54     3.46428E-37
55     2.56218E-38
56     1.83207E-39
57     1.26659E-40
58     8.46678E-42
59     5.47264E-43
60     3.4204E-44
Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Great answer to a very difficult problem. I think the key point is that it is not possible to calculate an exact answer, which is an important and useful insight.

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#### Scott A Wilson

##### Expertise

I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

##### Experience

Experience in the area; I have tutored people in the above areas of mathematics for over two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

Publications
My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

Education/Credentials
Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

Past/Present Clients
My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.