which one is worse in terms of probability












-1












$begingroup$


There are 2N white balls and N red balls (all balls are same except for the color), to put into K different boxes, such that every box contains 3N/k balls. We say event A happens, if any box has more than one half red balls.




  1. Mix 2N white balls and N red balls uniformly then put them into K boxes randomly;

  2. First, Put some red balls to K boxes equally, then mix 2N white balls and the rest red balls uniformly and lastly put them into K boxes randomly;


Q: which case has a higher probability of A?



Actually, we regard A as some "bad" case. Intuitively, the latter is more "uniform" so with less chance to have a "overflowd" box. I was trying to prove it formally. Here are my thinkings:



Using hypergeometry distribution, we write down the probability of negative A, so my target is to prove (Here mk is the red balls that put into boxes at very beginning in case 2, for convenience, I assume it is m times of k)



begin{equation}
frac{sumlimits_{substack{s_1+...+s_k leq N \ 0 le s_i le M/2}}{prod_{i=1}^k{C_M^{s_i}}}}{C_{3N}^N} leq frac{sumlimits_{substack{s_1+...+s_k leq N-mk \ 0 le s_i le M/2-k}}{prod_{i=1}^k{C_{M-m}^{s_i}}}}{C_{3N-mk}^{N-mk}}
end{equation}



I've tried several scale-down tricks, but none of the methods I know work.



Can anybody give me some idea? I feel this is a typical question, related materials is also a big help!






probability combinations generating-functions hypergeometric-function balls-in-bins







share|cite|improve this question











$endgroup$












  • $begingroup$
    You might get more positive response(s) if you show your attempts to figure this out with combinatorics logic.
    $endgroup$
    – poetasis
    Jan 5 at 12:24










  • $begingroup$
    by combinatorics logic, you mean I should write down some formula?
    $endgroup$
    – chuangmingjj
    Jan 5 at 13:26










  • $begingroup$
    I don't know if you need permutations, combinations, inclusion/exclusion but showing what you have tried always gets better responses. Even the right side of this screen shows related questions that may or may not provide insight into your problem. Good luck.
    $endgroup$
    – poetasis
    Jan 5 at 16:11
















-1












$begingroup$


There are 2N white balls and N red balls (all balls are same except for the color), to put into K different boxes, such that every box contains 3N/k balls. We say event A happens, if any box has more than one half red balls.




  1. Mix 2N white balls and N red balls uniformly then put them into K boxes randomly;

  2. First, Put some red balls to K boxes equally, then mix 2N white balls and the rest red balls uniformly and lastly put them into K boxes randomly;


Q: which case has a higher probability of A?



Actually, we regard A as some "bad" case. Intuitively, the latter is more "uniform" so with less chance to have a "overflowd" box. I was trying to prove it formally. Here are my thinkings:



Using hypergeometry distribution, we write down the probability of negative A, so my target is to prove (Here mk is the red balls that put into boxes at very beginning in case 2, for convenience, I assume it is m times of k)



begin{equation}
frac{sumlimits_{substack{s_1+...+s_k leq N \ 0 le s_i le M/2}}{prod_{i=1}^k{C_M^{s_i}}}}{C_{3N}^N} leq frac{sumlimits_{substack{s_1+...+s_k leq N-mk \ 0 le s_i le M/2-k}}{prod_{i=1}^k{C_{M-m}^{s_i}}}}{C_{3N-mk}^{N-mk}}
end{equation}



I've tried several scale-down tricks, but none of the methods I know work.



Can anybody give me some idea? I feel this is a typical question, related materials is also a big help!






probability combinations generating-functions hypergeometric-function balls-in-bins







share|cite|improve this question











$endgroup$












  • $begingroup$
    You might get more positive response(s) if you show your attempts to figure this out with combinatorics logic.
    $endgroup$
    – poetasis
    Jan 5 at 12:24










  • $begingroup$
    by combinatorics logic, you mean I should write down some formula?
    $endgroup$
    – chuangmingjj
    Jan 5 at 13:26










  • $begingroup$
    I don't know if you need permutations, combinations, inclusion/exclusion but showing what you have tried always gets better responses. Even the right side of this screen shows related questions that may or may not provide insight into your problem. Good luck.
    $endgroup$
    – poetasis
    Jan 5 at 16:11














-1












-1








-1


0



$begingroup$


There are 2N white balls and N red balls (all balls are same except for the color), to put into K different boxes, such that every box contains 3N/k balls. We say event A happens, if any box has more than one half red balls.




  1. Mix 2N white balls and N red balls uniformly then put them into K boxes randomly;

  2. First, Put some red balls to K boxes equally, then mix 2N white balls and the rest red balls uniformly and lastly put them into K boxes randomly;


Q: which case has a higher probability of A?



Actually, we regard A as some "bad" case. Intuitively, the latter is more "uniform" so with less chance to have a "overflowd" box. I was trying to prove it formally. Here are my thinkings:



Using hypergeometry distribution, we write down the probability of negative A, so my target is to prove (Here mk is the red balls that put into boxes at very beginning in case 2, for convenience, I assume it is m times of k)



begin{equation}
frac{sumlimits_{substack{s_1+...+s_k leq N \ 0 le s_i le M/2}}{prod_{i=1}^k{C_M^{s_i}}}}{C_{3N}^N} leq frac{sumlimits_{substack{s_1+...+s_k leq N-mk \ 0 le s_i le M/2-k}}{prod_{i=1}^k{C_{M-m}^{s_i}}}}{C_{3N-mk}^{N-mk}}
end{equation}



I've tried several scale-down tricks, but none of the methods I know work.



Can anybody give me some idea? I feel this is a typical question, related materials is also a big help!






probability combinations generating-functions hypergeometric-function balls-in-bins







share|cite|improve this question











$endgroup$




There are 2N white balls and N red balls (all balls are same except for the color), to put into K different boxes, such that every box contains 3N/k balls. We say event A happens, if any box has more than one half red balls.




  1. Mix 2N white balls and N red balls uniformly then put them into K boxes randomly;

  2. First, Put some red balls to K boxes equally, then mix 2N white balls and the rest red balls uniformly and lastly put them into K boxes randomly;


Q: which case has a higher probability of A?



Actually, we regard A as some "bad" case. Intuitively, the latter is more "uniform" so with less chance to have a "overflowd" box. I was trying to prove it formally. Here are my thinkings:



Using hypergeometry distribution, we write down the probability of negative A, so my target is to prove (Here mk is the red balls that put into boxes at very beginning in case 2, for convenience, I assume it is m times of k)



begin{equation}
frac{sumlimits_{substack{s_1+...+s_k leq N \ 0 le s_i le M/2}}{prod_{i=1}^k{C_M^{s_i}}}}{C_{3N}^N} leq frac{sumlimits_{substack{s_1+...+s_k leq N-mk \ 0 le s_i le M/2-k}}{prod_{i=1}^k{C_{M-m}^{s_i}}}}{C_{3N-mk}^{N-mk}}
end{equation}



I've tried several scale-down tricks, but none of the methods I know work.



Can anybody give me some idea? I feel this is a typical question, related materials is also a big help!






probability combinations generating-functions hypergeometric-function balls-in-bins




probability combinations generating-functions hypergeometric-function balls-in-bins






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 9 at 15:24







chuangmingjj

















asked Jan 5 at 8:05









chuangmingjjchuangmingjj

61




61












  • $begingroup$
    You might get more positive response(s) if you show your attempts to figure this out with combinatorics logic.
    $endgroup$
    – poetasis
    Jan 5 at 12:24










  • $begingroup$
    by combinatorics logic, you mean I should write down some formula?
    $endgroup$
    – chuangmingjj
    Jan 5 at 13:26










  • $begingroup$
    I don't know if you need permutations, combinations, inclusion/exclusion but showing what you have tried always gets better responses. Even the right side of this screen shows related questions that may or may not provide insight into your problem. Good luck.
    $endgroup$
    – poetasis
    Jan 5 at 16:11


















  • $begingroup$
    You might get more positive response(s) if you show your attempts to figure this out with combinatorics logic.
    $endgroup$
    – poetasis
    Jan 5 at 12:24










  • $begingroup$
    by combinatorics logic, you mean I should write down some formula?
    $endgroup$
    – chuangmingjj
    Jan 5 at 13:26










  • $begingroup$
    I don't know if you need permutations, combinations, inclusion/exclusion but showing what you have tried always gets better responses. Even the right side of this screen shows related questions that may or may not provide insight into your problem. Good luck.
    $endgroup$
    – poetasis
    Jan 5 at 16:11
















$begingroup$
You might get more positive response(s) if you show your attempts to figure this out with combinatorics logic.
$endgroup$
– poetasis
Jan 5 at 12:24




$begingroup$
You might get more positive response(s) if you show your attempts to figure this out with combinatorics logic.
$endgroup$
– poetasis
Jan 5 at 12:24












$begingroup$
by combinatorics logic, you mean I should write down some formula?
$endgroup$
– chuangmingjj
Jan 5 at 13:26




$begingroup$
by combinatorics logic, you mean I should write down some formula?
$endgroup$
– chuangmingjj
Jan 5 at 13:26












$begingroup$
I don't know if you need permutations, combinations, inclusion/exclusion but showing what you have tried always gets better responses. Even the right side of this screen shows related questions that may or may not provide insight into your problem. Good luck.
$endgroup$
– poetasis
Jan 5 at 16:11




$begingroup$
I don't know if you need permutations, combinations, inclusion/exclusion but showing what you have tried always gets better responses. Even the right side of this screen shows related questions that may or may not provide insight into your problem. Good luck.
$endgroup$
– poetasis
Jan 5 at 16:11










1 Answer
1






active

oldest

votes


















0












$begingroup$

Some thoughts too long for a comment. Let us consider the case where $N$ is quite large so the correlation between boxes becomes small. In case 1 the distribution of red and white balls in a box is Poisson. We can take case 2 to the extreme, distribute all the red balls evenly, and have exactly $frac NK$ red balls in each box. We can then sum over the Poisson distribution and find the chance that a particular bin causes event A. If $frac NK=1$ the chance one bin causes event A is about $0.183$ in case 1 and $0.135$ in case 2. If $frac NK=10$ the chances are $0.0258$ in case 1 and $0.0050$ in case 2. I suspect this is general-increasing the variability is good for the chance of unlikely things happening and distributing the red balls randomly increases the variability. Clearly this is not a proof, but it suggests the result.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    thanks for your thoughts! I have two questions. 1. Why it's Possion and what's the $lambda$ in Possion? As far as I know, binomial distribution is approximated by Possion if n is large, p is small, but $lambda=np$ is a constant. Here, my p is 1/3, that's not small. 2. How to get 0.183? Would you mind explaining a bit more
    $endgroup$
    – chuangmingjj
    Jan 10 at 2:53










  • $begingroup$
    It is Poisson if N is large because each ball has $1/N$ chance of falling in each bin. For the red balls $lambda=frac NK$ while for the white balls it is twice that
    $endgroup$
    – Ross Millikan
    Jan 10 at 3:28










  • $begingroup$
    I got $0.183$ in the pedestrian way. It is the chance (using Poisson) that there is one red and no whites plus the chance there is two reds and zero or one white plus...
    $endgroup$
    – Ross Millikan
    Jan 10 at 6:06











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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









0












$begingroup$

Some thoughts too long for a comment. Let us consider the case where $N$ is quite large so the correlation between boxes becomes small. In case 1 the distribution of red and white balls in a box is Poisson. We can take case 2 to the extreme, distribute all the red balls evenly, and have exactly $frac NK$ red balls in each box. We can then sum over the Poisson distribution and find the chance that a particular bin causes event A. If $frac NK=1$ the chance one bin causes event A is about $0.183$ in case 1 and $0.135$ in case 2. If $frac NK=10$ the chances are $0.0258$ in case 1 and $0.0050$ in case 2. I suspect this is general-increasing the variability is good for the chance of unlikely things happening and distributing the red balls randomly increases the variability. Clearly this is not a proof, but it suggests the result.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    thanks for your thoughts! I have two questions. 1. Why it's Possion and what's the $lambda$ in Possion? As far as I know, binomial distribution is approximated by Possion if n is large, p is small, but $lambda=np$ is a constant. Here, my p is 1/3, that's not small. 2. How to get 0.183? Would you mind explaining a bit more
    $endgroup$
    – chuangmingjj
    Jan 10 at 2:53










  • $begingroup$
    It is Poisson if N is large because each ball has $1/N$ chance of falling in each bin. For the red balls $lambda=frac NK$ while for the white balls it is twice that
    $endgroup$
    – Ross Millikan
    Jan 10 at 3:28










  • $begingroup$
    I got $0.183$ in the pedestrian way. It is the chance (using Poisson) that there is one red and no whites plus the chance there is two reds and zero or one white plus...
    $endgroup$
    – Ross Millikan
    Jan 10 at 6:06
















0












$begingroup$

Some thoughts too long for a comment. Let us consider the case where $N$ is quite large so the correlation between boxes becomes small. In case 1 the distribution of red and white balls in a box is Poisson. We can take case 2 to the extreme, distribute all the red balls evenly, and have exactly $frac NK$ red balls in each box. We can then sum over the Poisson distribution and find the chance that a particular bin causes event A. If $frac NK=1$ the chance one bin causes event A is about $0.183$ in case 1 and $0.135$ in case 2. If $frac NK=10$ the chances are $0.0258$ in case 1 and $0.0050$ in case 2. I suspect this is general-increasing the variability is good for the chance of unlikely things happening and distributing the red balls randomly increases the variability. Clearly this is not a proof, but it suggests the result.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    thanks for your thoughts! I have two questions. 1. Why it's Possion and what's the $lambda$ in Possion? As far as I know, binomial distribution is approximated by Possion if n is large, p is small, but $lambda=np$ is a constant. Here, my p is 1/3, that's not small. 2. How to get 0.183? Would you mind explaining a bit more
    $endgroup$
    – chuangmingjj
    Jan 10 at 2:53










  • $begingroup$
    It is Poisson if N is large because each ball has $1/N$ chance of falling in each bin. For the red balls $lambda=frac NK$ while for the white balls it is twice that
    $endgroup$
    – Ross Millikan
    Jan 10 at 3:28










  • $begingroup$
    I got $0.183$ in the pedestrian way. It is the chance (using Poisson) that there is one red and no whites plus the chance there is two reds and zero or one white plus...
    $endgroup$
    – Ross Millikan
    Jan 10 at 6:06














0












0








0





$begingroup$

Some thoughts too long for a comment. Let us consider the case where $N$ is quite large so the correlation between boxes becomes small. In case 1 the distribution of red and white balls in a box is Poisson. We can take case 2 to the extreme, distribute all the red balls evenly, and have exactly $frac NK$ red balls in each box. We can then sum over the Poisson distribution and find the chance that a particular bin causes event A. If $frac NK=1$ the chance one bin causes event A is about $0.183$ in case 1 and $0.135$ in case 2. If $frac NK=10$ the chances are $0.0258$ in case 1 and $0.0050$ in case 2. I suspect this is general-increasing the variability is good for the chance of unlikely things happening and distributing the red balls randomly increases the variability. Clearly this is not a proof, but it suggests the result.






share|cite|improve this answer









$endgroup$



Some thoughts too long for a comment. Let us consider the case where $N$ is quite large so the correlation between boxes becomes small. In case 1 the distribution of red and white balls in a box is Poisson. We can take case 2 to the extreme, distribute all the red balls evenly, and have exactly $frac NK$ red balls in each box. We can then sum over the Poisson distribution and find the chance that a particular bin causes event A. If $frac NK=1$ the chance one bin causes event A is about $0.183$ in case 1 and $0.135$ in case 2. If $frac NK=10$ the chances are $0.0258$ in case 1 and $0.0050$ in case 2. I suspect this is general-increasing the variability is good for the chance of unlikely things happening and distributing the red balls randomly increases the variability. Clearly this is not a proof, but it suggests the result.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 9 at 15:58









Ross MillikanRoss Millikan

299k24200374




299k24200374












  • $begingroup$
    thanks for your thoughts! I have two questions. 1. Why it's Possion and what's the $lambda$ in Possion? As far as I know, binomial distribution is approximated by Possion if n is large, p is small, but $lambda=np$ is a constant. Here, my p is 1/3, that's not small. 2. How to get 0.183? Would you mind explaining a bit more
    $endgroup$
    – chuangmingjj
    Jan 10 at 2:53










  • $begingroup$
    It is Poisson if N is large because each ball has $1/N$ chance of falling in each bin. For the red balls $lambda=frac NK$ while for the white balls it is twice that
    $endgroup$
    – Ross Millikan
    Jan 10 at 3:28










  • $begingroup$
    I got $0.183$ in the pedestrian way. It is the chance (using Poisson) that there is one red and no whites plus the chance there is two reds and zero or one white plus...
    $endgroup$
    – Ross Millikan
    Jan 10 at 6:06


















  • $begingroup$
    thanks for your thoughts! I have two questions. 1. Why it's Possion and what's the $lambda$ in Possion? As far as I know, binomial distribution is approximated by Possion if n is large, p is small, but $lambda=np$ is a constant. Here, my p is 1/3, that's not small. 2. How to get 0.183? Would you mind explaining a bit more
    $endgroup$
    – chuangmingjj
    Jan 10 at 2:53










  • $begingroup$
    It is Poisson if N is large because each ball has $1/N$ chance of falling in each bin. For the red balls $lambda=frac NK$ while for the white balls it is twice that
    $endgroup$
    – Ross Millikan
    Jan 10 at 3:28










  • $begingroup$
    I got $0.183$ in the pedestrian way. It is the chance (using Poisson) that there is one red and no whites plus the chance there is two reds and zero or one white plus...
    $endgroup$
    – Ross Millikan
    Jan 10 at 6:06
















$begingroup$
thanks for your thoughts! I have two questions. 1. Why it's Possion and what's the $lambda$ in Possion? As far as I know, binomial distribution is approximated by Possion if n is large, p is small, but $lambda=np$ is a constant. Here, my p is 1/3, that's not small. 2. How to get 0.183? Would you mind explaining a bit more
$endgroup$
– chuangmingjj
Jan 10 at 2:53




$begingroup$
thanks for your thoughts! I have two questions. 1. Why it's Possion and what's the $lambda$ in Possion? As far as I know, binomial distribution is approximated by Possion if n is large, p is small, but $lambda=np$ is a constant. Here, my p is 1/3, that's not small. 2. How to get 0.183? Would you mind explaining a bit more
$endgroup$
– chuangmingjj
Jan 10 at 2:53












$begingroup$
It is Poisson if N is large because each ball has $1/N$ chance of falling in each bin. For the red balls $lambda=frac NK$ while for the white balls it is twice that
$endgroup$
– Ross Millikan
Jan 10 at 3:28




$begingroup$
It is Poisson if N is large because each ball has $1/N$ chance of falling in each bin. For the red balls $lambda=frac NK$ while for the white balls it is twice that
$endgroup$
– Ross Millikan
Jan 10 at 3:28












$begingroup$
I got $0.183$ in the pedestrian way. It is the chance (using Poisson) that there is one red and no whites plus the chance there is two reds and zero or one white plus...
$endgroup$
– Ross Millikan
Jan 10 at 6:06




$begingroup$
I got $0.183$ in the pedestrian way. It is the chance (using Poisson) that there is one red and no whites plus the chance there is two reds and zero or one white plus...
$endgroup$
– Ross Millikan
Jan 10 at 6:06


















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.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
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