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June 26th, 2019, 08:30 AM  #1 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 650 Thanks: 86  Probability With the Numbers From 1 to 4 Four Times Each
There are 16 balls, with each number from 1 to 4 four times. Balls are drawn without replacement. How many balls need to be drawn to make at least a 50 percent chance that any one number will have all four balls drawn?

June 27th, 2019, 01:06 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,494 Thanks: 1369 
Well... I've got a method that works but it's pretty brute force. For each $n$ you come up with a 4 digit integer partition of $n$ such that each digit is at most 3 Then evaluate the probabilities of each of those and sum them. Subtract the result from 1. This is the probability that at least 1 of the labels appears 4 times in the $n$ selections. $p(\{1,2,\dots,16\}) = \left\{0,0,0,\frac{1}{455},\frac{1}{91},\frac{3}{9 1},\frac{1}{13},\frac{329}{2145},\frac{3}{11}, \frac{63}{143},\frac{59}{91},\frac{391}{455},1,1,1 ,1 \right\}$ $n=10 \Rightarrow p(10)=\dfrac{59}{91}$ is the first to be greater than 50%. These answers do accord with the totally brute force method of having Mathematica gen up all the possible arrangements and counting them, so I have pretty high confidence in them. Whether there's a more elegant solution I don't know. 
June 28th, 2019, 05:33 PM  #3 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 650 Thanks: 86 
For the probabilities other than 0 and 1, I gave them the least common denominator of 15,015. I'm wondering if the numerators can be expressed in n P r form or otherwise show a pattern. 1/455 = 33/15,015 1/91 = 165/15,015 3/91 = 495/15,015 1/13 = 1,155/15,015 329/2145 = 2,303/15,015 3/11 = 2,095/15,015 63/143 = 6,615/15,015 59/91 = 9,735/15,015 391/455 = 12,903/15,015 
June 28th, 2019, 05:38 PM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,494 Thanks: 1369 
I looked up the numerator sequence when put over a common denominator on OEIS. The sequence was embedded in a half dozen sequences but didn't seem to have it's own. 

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