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Why is the number of possible events given by number of subsets of sample space?

I want to understand this using this question:

A coin is tossed $n$ times. What is the number of elements in its sample space? Also, write the number of all possible events.

Number of outcomes = $2 \times 2 \times 2...n \mathbb{times}$= $2^n$

Now, number of outcomes denotes the number of elements in the sample space.

Let's say $n = 2$, then Sample Space S = $\{HH,TT, HT, TH\}$ . Now this set's subsets are : $\phi$, $\{HH, TT\},\{HH\},\{TT\},\{HH\},\{TH\}, \{HT\}, \{HH, TT, HT\} ...$ Also, look at the subset with 3 possibilities, how can that be a possibility when the coin has just been tossed twice?

I just can't get how the number of subsets can denote the possible number of events using this example.

Tl, dr: How is the set of all events, the power set (i.e. set of all subsets) of the sample space?

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    $\begingroup$ So are you basically asking why the set of all events is the power set (i.e. set of all subsets) of the sample space? $\endgroup$ – M. Winter Dec 18 '17 at 12:06
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    $\begingroup$ Yes @M.Winter... $\endgroup$ – Archer Dec 18 '17 at 12:07
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    $\begingroup$ Added that to the question. $\endgroup$ – Archer Dec 18 '17 at 12:29
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Maybe you are lacking a good intuition on what an event is.

An event is not some specific outcome of an experiment, but more like a property which an outcome either does have or not have.

Example. If you throw $n=4$ coins, then a result of your experiment is a sequence on four many heads or tails. But an event could be "there are at least two heads". Now, an outcome does have this property or it does not.

Example. Here is a property whith a different description than the last: "at least half of the outcomes are heads". Despite the different words, both events are actually the same: there are at least two heads if and only of at least the half of the outcomes is head.

So we can indeed give properties by words as I did above, but we may hide similarities. A better way to give a property is by listing all the possible outcomes which have this property. For both above wordings we obtain

\begin{align} &\mathrm{HHHH},\quad \mathrm{HHHT},\quad \mathrm{HHTH},\quad \mathrm{HTHH},\\ &\mathrm{THHH},\quad \mathrm{HHTT},\quad \mathrm{HTHT},\quad \mathrm{THHT},\\ &\mathrm{HTTH},\quad \mathrm{THTH},\quad \mathrm{TTHH}. \end{align}

In other words, we can associate a property with the set of all outcomes which have this property, and this is a language and formulation independent description. This set is a subset of the state space because it does not necessarily contain all states, but maybe just some.

But we can also think the other way. Any possible subset of the statespace can be seen as the set of outcomes which have a specific property. We might not have a formulation in words for this property, but we could find one if we search for it. It is just not necessary. The subset is enough.

Example. The subset $$\{\mathrm{HHHH},\mathrm{HHTT}\}$$ can be seen as the property "there are either no tails, or the third and fourth throw are tails". The empty set $\varnothing$ is an event too. It is a property which no sequence of four coin tosses has, e.g. "there are at least seven heads".

So all subsets are valid event descriptions. Therefore, the set of all events can be associated with the set of all subsets $-$ the powerset. And it is a basic mathematical fact that if a set $X$ has $k=|X|$ many elements, then the powerset has $2^k$ many elements.

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