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As title. Suppose that the sample space $S$ of some experiment is finite. Show that the collection of all subsets of $S$ satisfies the three conditions required to be called the collection of events.

So sample space $S$ is finite, with $N$ possible outcomes, where $N$ is a positive integer. Thus $S$ has the form $S = \{w_1, ..., w_N\}$, where $w_i$ represents the $i$-th possible outcome, for $i \in \{1, ..., N\}$.

This sample space contains $2^N$ possible events like a power set.

We can construct a probability measure by defining probabilities $p_i$ for each particular outcome $w_i \in S$.

$P[w_i] = p_i, \forall i \in \{1, ..., N\}$

I was thinking of defining the probability measure to all events $A \subseteq S$ by defining $P(\emptyset)=0$, and $P(A)$ for each non-empty subset $A \subseteq S$ as follows:

$P(A) = \sum_{w_i \in A} P(w_i)$

where the summation on the right-hand side represents a sum over each of the elements $w_i$ in the set $A$.

My question is, am I on the right track, if so, how do I show that this definition satisfies the three axioms of probability? That is,

  1. $S$ must be an event.

  2. If $a \in S$ is an event, then so is $a^c$.

  3. If $a_1, ..., a_n$ is a countable collection of events, then $\bigcup_{i=1}^{\infty} a_i$ is also an event.

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    $\begingroup$ It is not necessary to assign probabilities etc. for the task you have to do. You need to only verify that the collection of all subsets of the sample space (there are $2^n$ such subsets) includes the sample space, complements of all subsets, and countable unions of subsets. So start! Can $S$ be regarded as a subset of itself and thus is included in the collection of all subsets of $S$? If $A$ is a subset of $S$, is $A^c = S-A$ a subset of $S$ also? and so on. $\endgroup$ – Dilip Sarwate Sep 2 '14 at 1:57
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Much of the post is not on the right track. The part that is on the right track is the list of three conditions. You need to show that the collection of all subsets of a (finite) set satisfies these conditions. The verifications will be in each case very short.

For example, is $S$ in our collection? Sure, every subset of $S$ is in our collection, and $S$ is a subset of itself.

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  • $\begingroup$ I think I'm confused with the difference of $S$ and the collection of all subsets of $S$. Say if $S = \{1,2\}$, then the collection of all subsets of $S$ will be $\{\emptyset\}, \{1\}, \{2\}, \{1,2\}$ right? If I let $A = \{1\}$, then what does $A^c$ look like? $A^c=\{2\}$ or $A^c=\{\emptyset\}, \{2\}, \{1,2\}$? @Dilip Sarwate as well. Thanks you guys in advance! $\endgroup$ – PandaMan Sep 2 '14 at 2:13
  • $\begingroup$ If $A=\{1\}$ then $A^c=\{2\}$. We want the set of all elements of $S$ that are not in $A$. $\endgroup$ – André Nicolas Sep 2 '14 at 2:16

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