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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

worry about.

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

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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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