4.1: Outcomes and events
Since life is uncertain, we don’t know for sure what is going to happen. But let’s start by assuming we know what things might happen. Something that might happen is called an outcome . You can think of this as the result of an experiment if you want to, although normally we won’t be talking about outcomes that we have explicitly manipulated and measured via scientific means. It’s more like we’re just curious how some particular happening is going to turn out, and we’ve identified the different ways it can turn out and called them outcomes.
Now we’ve been using the symbol \(\Omega\) to refer to “the domain of discourse" or “the universal set" or “all the stuff we’re talking about." We’re going to give it yet another name now: the sample space . \(\Omega\) , the sample space, is simply the set of all possible outcomes. Any particular outcome — call it \(O\) — is an element of this set, just like in chapter 1 every conceivable element was a member of the domain of discourse.
If a woman is about to have a baby, we might define \(\Omega\) as { boy, girl }. Any particular outcome \(o\) is either boy or girl (not both), but both outcomes are in the sample space, because both are possible. If we roll a die, we’d define \(\Omega\) as { 1, 2, 3, 4, 5, 6 }. If we’re interested in motor vehicle safety, we might define \(\Omega\) for a particular road trip as { safe, accident }. The outcomes don’t have to be equally likely, an important point we’ll return to soon.
In probability, we define an event as a subset of the sample space . In other words, an event is a group of related outcomes (though an event might contain just one outcome, or even zero). I always thought this was a funny definition for the word “event": it’s not the first thing that word brings to mind. But it turns out to be a useful concept, because sometimes we’re not interested in any particular outcome necessarily, but rather in whether the outcome — whatever it is — has a certain property. For instance, suppose at the start of some game, my opponent and I each roll the die, agreeing that the highest roller gets to go first. Suppose he rolls a 2. Now it’s my turn. The \(\Omega\) for my die roll is of course { 1, 2, 3, 4, 5, 6 }. But in this case, it doesn’t necessarily matter what my specific outcome is; only whether I beat a 2. So I could define the event \(M\) (for “me first") to be the set { 3, 4, 5, 6 }. I could define the event \(H\) (“him first") to be the set { 1 } (notice \(H\) is still a set, even though it has only one element.) Then I could define the event \(T\) (“tie") as the set { 2 }. I’ve now effectively collapsed a larger set of outcomes into only the groups of outcomes I’m interested in. Now I’m all ready to reason about the likelihood that each of these events actually occurs.
By the way, “the set of all outcomes" is simply \(\Omega\) , since an outcome is an element of \(\Omega\) . But an event is a subset of \(\Omega\) , not a single element. What, then, is “the set of all events?" If you think it through, you’ll realize that it’s \(\mathbb{P}(\Omega)\) (the power set of the sample space). Put another way, when defining an event, I can choose any subset of the possible outcomes, and so I can choose any set from \(\mathbb{P}(\Omega)\) .