# A simple fact about sets

Out of elementary events one can get

possible outcomes. Where is an event that contains elementary events. Take set

with the size as the only characteristic . Then it power set

contains elements. event for one element each, . Then events with two element, . Finally, event for one with all elements, . 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 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 elements of two type.

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

Instead of computing that manually one can use this asymptotic equality

Thus

Which means

Simple algebra yields

The result fascinates me. The graph visualizes data from a real experiment where a pack is split equally times and is a cumulated sum if exactly 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

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