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public:calculating_the_size_of_a_set_by_observing_the_proportionality_change_of_its_disjoint_subsets [2019/04/16 10:49] – [Background] fangfufupublic:calculating_the_size_of_a_set_by_observing_the_proportionality_change_of_its_disjoint_subsets [2019/04/17 02:08] (current) – [Formal Problem statement] fangfufu
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 So the questions are: how many people actually voted in the poll? How many people voted for each option?  So the questions are: how many people actually voted in the poll? How many people voted for each option? 
  
-After presenting my original solution to [[public:the_number_of_people_voted_in_an_instagram_poll|Instagram polling]] problem to Cosmin, [[https://scholar.google.co.uk/citations?user=S7UZ6MAAAAAJ&hl=en|Cosmin Gorgovan]], he said: +After presenting my original solution to [[public:the_number_of_people_voted_in_an_instagram_poll|Instagram polling]] problem to [[https://scholar.google.co.uk/citations?user=S7UZ6MAAAAAJ&hl=en|Cosmin Gorgovan]], he said: 
 <blockquote> <blockquote>
 If you make observations before and after one vote, you can directly calculate the total number of votes from the weight of that one vote. If you make observations before and after one vote, you can directly calculate the total number of votes from the weight of that one vote.
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 We can consider everyone who voted in an Instagram poll as a set, and the two options are disjoint subsets of the superset.  We can consider everyone who voted in an Instagram poll as a set, and the two options are disjoint subsets of the superset. 
  
-===== Formal Problem statement ===== +===== Formal problem statement ===== 
-Set $A$ consists of disjoint subsets $a_1, a_2, ..., a_n$. Although we do not know the cardinality of set $A$ (denoted by $|A|$) and the cardinality of each of the subset, we do know the proportion of each subset in terms of set $A$, that is we know $\frac{|a_1|}{|A|}, \frac{|a_2|}{|A|}, ... \frac{|a_n|}{|A|}$. We are allowed to add $n$ elements into subsets of $A$, and observe the changes in proportionality of the subsets. What is the cardinality of set $A$ before elements were added to its subsets?+Set $A$ consists of disjoint subsets $a_1, a_2, ..., a_n$. Although we do not know the cardinality of set $A$ (denoted by $|A|$) and the cardinality of each of the subset, we do know the proportion of each subset in terms of set $A$, that is we know $\frac{|a_1|}{|A|}, \frac{|a_2|}{|A|}, ... \frac{|a_n|}{|A|}$. We are allowed to add $n$ elements into subset $a_1$, and observe its change in proportionality. What is the cardinality of set $A$ before elements were added to $a_1$?
  
 ===== Solution ===== ===== Solution =====
-This problem can be solved by modelling the process of adding $n$ elements to subset $a_1$. We denote the proportionality of $a_1$ as $\alpha_1$, that is $\alpha_1 = \frac{|a_1|}{|A|}$, and the change in proportionality of $a_1$ using $\delta_{a_1}$+This problem can be solved by modelling the process of adding $n$ elements to subset $a_1$. We denote the proportionality of $a_1$ as $\alpha_1$, that is $\alpha_1 = \frac{|a_1|}{|A|}$, and the change in proportionality of $a_1$ as $\delta_{a_1}$
  
 We begin by writing down the process that leads to the proportionality change. We add $n$ elements into subset $a_1$  We begin by writing down the process that leads to the proportionality change. We add $n$ elements into subset $a_1$ 
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 By dividing the above equation by $|A|$ and rearrangement, we obtain: By dividing the above equation by $|A|$ and rearrangement, we obtain:
-$$ |A|  = \frac{-n(\delta_{a_1} - 1 + \alpha_1)}{\delta_{a_1}} $$+$$ |A|  = \frac{n(1 - \alpha_1 - \delta_{a_1})}{\delta_{a_1}} $$
  
 You know everything in the right hand side of the equation, so solving $|A|$ is very easy.  You know everything in the right hand side of the equation, so solving $|A|$ is very easy. 
  
public/calculating_the_size_of_a_set_by_observing_the_proportionality_change_of_its_disjoint_subsets.1555411742.txt.gz · Last modified: 2019/04/16 10:49 by fangfufu