A simple fact about sets

Out of n elementary events one can get

\sum_{m=1}^{n} C_{n}^{m} = 2^n - 1

possible outcomes. Where C_{n}^{m} is an event that contains m elementary events. Take set

\{ a,b,c\}

with the size as the only characteristic n=3. Then it power set

\{ \{ a\} ,\{ b\} ,\{ c\} ,\{ a,b\} ,\{ a,c\} ,\{ b,c\} ,\{ a,b,c\} ,\{ \emptyset \} \}

contains {2^3} = 8 elements. 3 event for one element each, C_{3}^{1}. Then 3 events with two element, C_{3}^{2}. Finally, 1 event for one with all elements, C_{3}^{1}. A emply set is an impossible event.

I personally think that this simple fact is amazing, but some would say it is kinda boring. Here is an interesting question for those.

A pack of cards that has 36 cards is randomly split equally into halves. What is the probability that halves have equal amount black and red cards?

This is just another set with 36 elements of two type.

p = \frac{{C_{18}^9 \times C_{18}^9}}{{C_{36}^{18}}} = \frac{{{{(18!)}^4}}}{{36!{{(9!)}^4}}}

The denominator indicates all possible equally likely ways the pack can be split.

Instead of computing that manually one can use this asymptotic equality

n!\ \approx \sqrt {2\pi n} \cdot {n^n}{e^{ - n}}


18!\ \approx {18^{18}}{e^{ - 18}}\sqrt {2\pi \cdot 18}

9!\ \approx {9^9}{e^{ - 9}}\sqrt {2\pi \cdot 9}

36!\ \approx {36^{36}} \cdot {e^{ - 36}}\sqrt {2\pi \cdot 36}

Which means

p \approx \frac{{{{(\sqrt {2\pi \cdot 18} \cdot {{18}^{18}} \cdot {e^{ - 18}})}^4}}}{{\sqrt {2\pi \cdot 36} \cdot {{36}^{36}} \cdot {e^{ - 36}}{{(\sqrt {2\pi \cdot 9} \cdot {9^9} \cdot {e^{ - 9}})}^4}}}

Simple algebra yields

p \approx \frac{2}{{\sqrt {18\pi } }} \approx \frac{4}{{15}} \approx 0.26

The result fascinates me. The graph visualizes data from a real experiment where a pack is split equally 100 times and \mu is a cumulated sum if exactly 9 red cards are observed in on of the halves. What is crazy is that we were able to see the results of this experiments without doing any experiments, by simply reasoning mathematically about things.

More on this topic: Гнеденко-1988 


I'd like to know your thoughts about it

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