public:calculating_the_size_of_a_set_by_observing_the_proportionality_change_of_its_disjoint_subsets
Differences
This shows you the differences between two versions of the page.
Both sides previous revisionPrevious revisionNext revision | Previous revision | ||
public:calculating_the_size_of_a_set_by_observing_the_proportionality_change_of_its_disjoint_subsets [2019/04/16 10:33] – fangfufu | public: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 | ||
---|---|---|---|
Line 1: | Line 1: | ||
====== Calculating the size of a set by observing the proportionality change of its disjoint subsets ====== | ====== Calculating the size of a set by observing the proportionality change of its disjoint subsets ====== | ||
===== Background ===== | ===== Background ===== | ||
- | After presenting my solution to [[public: | + | Instagram provides a polling feature, which allows the user to ask a question with two answers to the audiences. Once an audience pick one of the two answers, the percentage of users who pick each answer is displayed. |
+ | |||
+ | So the questions are: how many people actually voted in the poll? How many people voted for each option? | ||
+ | |||
+ | After presenting my original | ||
< | < | ||
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. | ||
Line 9: | Line 13: | ||
I thought he made a really good point. I thought this problem is actually mathematically interesting, | I thought he made a really good point. I thought this problem is actually mathematically interesting, | ||
- | ===== Formal | + | We can consider everyone who voted in an Instagram poll as a set, and the two options are disjoint subsets of the superset. |
- | 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|}, | + | |
+ | ===== Formal | ||
+ | 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|}, | ||
===== 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 $\alpha_1 = \frac{|a_1|}{|A|}$, | + | 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 $\alpha_1 = \frac{|a_1|}{|A|}$, |
- | We begin by writing down the process that leads to the proportionality change: | + | We begin by writing down the process that leads to the proportionality change. We add $n$ elements into subset $a_1$ |
$$ \frac{|a_1| + n }{|A| + n} - \frac{|a_1|}{|A|} = \delta_{a_1} $$ | $$ \frac{|a_1| + n }{|A| + n} - \frac{|a_1|}{|A|} = \delta_{a_1} $$ | ||
- | By multiplying the equation with the denominators in the right hand side, we obtain: | + | By multiplying the equation with the denominators in the left hand side, we obtain: |
$$ (|a_1| + n )|A| - |a_1|(|A| + n) = \delta_{a_1}(|A| + n)(|A|)$$ | $$ (|a_1| + n )|A| - |a_1|(|A| + n) = \delta_{a_1}(|A| + n)(|A|)$$ | ||
Line 38: | Line 44: | ||
By dividing the above equation by $|A|$ and rearrangement, | By dividing the above equation by $|A|$ and rearrangement, | ||
- | $$ |A| = \frac{-n(\delta_{a_1} | + | $$ |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. | ||
public/calculating_the_size_of_a_set_by_observing_the_proportionality_change_of_its_disjoint_subsets.1555410832.txt.gz · Last modified: 2019/04/16 10:33 by fangfufu