which one is worse in terms of probability
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.
- Mix 2N white balls and N red balls uniformly then put them into K boxes randomly;
- 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 thanked!
probability combinations generating-functions hypergeometric-function balls-in-bins
asked 2 days ago
chuangmingjj
61
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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.
- Mix 2N white balls and N red balls uniformly then put them into K boxes randomly;
- 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 thanked!
probability combinations generating-functions hypergeometric-function balls-in-bins
asked 2 days ago
chuangmingjj
61
New contributor
chuangmingjj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
You might get more positive response(s) if you show your attempts to figure this out with combinatorics logic.
– poetasis
yesterday
by combinatorics logic, you mean I should write down some formula?
– chuangmingjj
yesterday
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.
– poetasis
yesterday
add a comment |
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.
- Mix 2N white balls and N red balls uniformly then put them into K boxes randomly;
- 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 thanked!
probability combinations generating-functions hypergeometric-function balls-in-bins
asked 2 days ago
chuangmingjj
61
New contributor
chuangmingjj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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.
- Mix 2N white balls and N red balls uniformly then put them into K boxes randomly;
- 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 thanked!
probability combinations generating-functions hypergeometric-function balls-in-bins
probability combinations generating-functions hypergeometric-function balls-in-bins
asked 2 days ago
chuangmingjj
61
New contributor
chuangmingjj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
chuangmingjj
61
New contributor
chuangmingjj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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edited 18 hours ago
asked 2 days ago
chuangmingjj
61
New contributor
chuangmingjj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 2 days ago
chuangmingjj
61
asked 2 days ago
chuangmingjj
61
61
New contributor
chuangmingjj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
chuangmingjj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
chuangmingjj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
You might get more positive response(s) if you show your attempts to figure this out with combinatorics logic.
– poetasis
yesterday
by combinatorics logic, you mean I should write down some formula?
– chuangmingjj
yesterday
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.
– poetasis
yesterday
add a comment |
You might get more positive response(s) if you show your attempts to figure this out with combinatorics logic.
– poetasis
yesterday
by combinatorics logic, you mean I should write down some formula?
– chuangmingjj
yesterday
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.
– poetasis
yesterday
You might get more positive response(s) if you show your attempts to figure this out with combinatorics logic.
– poetasis
yesterday
You might get more positive response(s) if you show your attempts to figure this out with combinatorics logic.
– poetasis
yesterday
by combinatorics logic, you mean I should write down some formula?
– chuangmingjj
yesterday
by combinatorics logic, you mean I should write down some formula?
– chuangmingjj
yesterday
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.
– poetasis
yesterday
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.
– poetasis
yesterday
add a comment |
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You might get more positive response(s) if you show your attempts to figure this out with combinatorics logic.
– poetasis
yesterday
by combinatorics logic, you mean I should write down some formula?
– chuangmingjj
yesterday
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.
– poetasis
yesterday