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I've just started Wasserman's All of Statistics and he starts by saying:

"The sample space $\Omega$, is the set of possible outcomes of an experiment. Points $\omega$ in $\Omega$ are called sample outcomes or realizations. Events are subsets of $\Omega$."

So sample outcomes are just the one-element subsets of $\Omega$, while events are all subsets of $\Omega$?

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A little bit different: the sample outcomes, also called "elementary events", are the points of $\Omega$, the one-element subsets of $\Omega$ are also events (that is, in elementary probability, with finite or countable sample space $\Omega$, they are events, in the more general case, they might or might no be events.

THE REST IS TECHNICAL AND COULD BE IGNORED BY MANY

That is because, in a formal definition of probability space as a triple $(\Omega, \mathcal{B}, P)$, the events are the elements in $\mathcal{B}$, a sigma-algebra of subsets of the sample space $\Omega$. Those subsets in the sigma-algebra are given probabilities by the probability measure $P$, and those are the events. Other subsets are not given any probability, and are not events. In most practical cases, the one-point subsets will be events.

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    $\begingroup$ The singletons (one-element subsets) might or might not be events. $\endgroup$
    – whuber
    Aug 26 '15 at 15:30
  • $\begingroup$ Could you elaborate, then, on what "points" are and joe to know whether they constitute events? $\endgroup$
    – anon_swe
    Aug 26 '15 at 19:32
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    $\begingroup$ @Antoni BTW, I see the source of the confusion: Wikipedia uses "event" inconsistently. The page you reference assumes an event could be any subset of $\Omega$ and states that some events just have undefined probabilities (which is a strange thing to assert). However, for the definition of "event" it references en.wikipedia.org/wiki/Event_%28probability_theory%29, where it states "an event is ... a subset of the sample space ... to which a probability is assigned." This latter sense is, I believe, the conventional one. $\endgroup$
    – whuber
    Aug 27 '15 at 17:03
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    $\begingroup$ @whuber Thank you for your detailed explanation. Can I conclude that a singleton is a subset of $\Omega$ which might be an event if there is a probability assigned to it, or otherwise still be a subset, but not an event? It's hard to come up with an example with dice or cards... $\endgroup$ Aug 27 '15 at 17:37
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    $\begingroup$ @Antoni That's correct. For a cards example you could (say) stipulate that the only events you want to ascribe probabilities to are the suits. All events would have 0, 13, 26, 39, or 52 elements, corresponding to which suits they contain. Although this is artificial, it illustrates the distinction between an event and a subset. Questions like "what is the chance the card is red" would have well-defined answers, but questions like "what is the chance the card is the queen of hearts" would not. None of these events are singletons. $\endgroup$
    – whuber
    Aug 27 '15 at 18:30
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[Tentative answer. Thank you very much to William Huber for his patient guidance as apparent in the comments.]

The probability space can be thought as a construct based on three decision levels:

Decision 1: The Random Experiment

The experiment can be simply observing and counting something we have no control over, or tossing a die in the air once, or an infinite number of times. A random experiment fulfills two conditions: 1. It has more than one possible result (or outcome); and 2. The results cannot be predicted in advance.

Decision 2: The Outcome

We can focus in any of the virtually infinite aspect of the "experiment". For instance if we are tossing a die, we can measure how high above the table it flies; how far away it travels; in what quadrant of the table it lands; how many rolls it takes before settling; or the more obvious outcome: what side is facing up when it stops moving (heads or tails).

After deciding what outcome we care about, the sample space will be immediately defined as the set of all possible elementary outcomes. These outcomes or elements of the sample space are denoted by the letter $\omega$.

If we look at the quadrant of the table where the die lands, the sample space will be $\Omega =\small\{LUQ, RUQ, LLQ, RLQ\}$ (left-upper quadrant, right-upper quadrant,...), whereas if we record the most typical outcome (the number facing up), the sample space will be $\Omega= \small\{1, 2, 3, 4, 5, 6\}$. $\small 1$, $\small 2$,... $\small 6$ being the elementary outcomes constituting the sample space.

Any grouping of elements of $\Omega$ corresponds to a subset, for instance $A_1 =\small \{1, 4\}$. Singletons are one-element subsets within the set: $A_2 = \small \{1\}$ and $A_3 = \small \{4\}$.

If the experiment consists of tossing three coins and we are interested in checking the Heads v Tails dichotomous outcome, the sample space will be $\Omega= \small\{HHH,HHT,HTH,THH,TTT,TTH,THT,HTT\} =\{H,T\}^3$ containing $2^3$ elementary outcomes. These are not $\small H$ or $\small T$, but rather $\small HHH$, $\small HHT$, etc.

Decision 3: The Event

Of all the possible subsets in the sample space $\Omega$ (imagine circles around groups of elementary outcomes) only some will be interesting to us in terms of assigning probabilities to them. It is the fact that we assign a probability measure to them that makes them events. Events can be simple (singletons) if they correspond to subsets with just one element from the sample space; or compound if they encompass more than one point in $\Omega$.

For instance, the event defined as "only one $\small T$" can be expressed as $A = \small \{HHT, HTH, THH\}$. This is a subset of the sample space of the experiment throwing three coins, but not any subset - it is the subset we want to assign a probability measure to.

An event has occurred when the outcome (denoted as $\omega$ to remind us that $\omega \in \Omega$) is also contained in the subset $A$, that is, $\omega \in A$.

The original question was:

"So sample outcomes are just the one-element subsets of $\Omega$, while events are all subsets of $\Omega$?"

The first part of the question sounds at first correct, except for the mathematical inconsistency pointed in the comments: an outcome ($\omega$) is not a subset ($\{\omega\}$). Perhaps the correct terminology is "the sample outcomes are contained in one-element subsets or singletons" (?).

Not all subsets of $\Omega$ are events: Events are only those subsets of $\Omega$ we happen to be interested in. They are the collections of subsets of $\Omega$ within the $\sigma$-algebra ($\mathcal{B}$), forming together with the sample space ($\Omega$) a measurable space. Once we assign probabilities to these collections of subsets of $\Omega$ we complete the probability space $(\Omega, \mathcal{B}, P)$.

On the other hand, all events are subsets of $\Omega$.

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  • $\begingroup$ It's very good now, Antoni, except for one small thing: one paragraph after stating you will use "$\omega$" to designate outcomes, you use it instead to designate subsets. I suspect you intended to use something like $A$ in that example rather than $\omega$. $\endgroup$
    – whuber
    Sep 18 '15 at 14:09

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