Statistics runs on probability underneath. A hypothesis test is a probability argument. A confidence interval is built from one too. Every sampling method in the next chapter is a rule for choosing outcomes from a population. This section is a fast refresher, not new content. It reviews the vocabulary that lets all of that be stated precisely.
Quick review
Practice from earlier coursework, pulled in for this topic.
From Probability
Which of these is guaranteed true for ANY two events A and B?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Not guaranteed — if B ⊆ A, the union equals A exactly, with no increase.
A ∩ B can absolutely be empty — that's exactly what 'mutually exclusive' means.
Two events sharing no outcomes at all (mutually exclusive) is a normal, common case.
Why
The intersection is always a subset of the union (every outcome in both A and B is also in A or B), so it can never have more outcomes.
Test each choice against a mutually exclusive pair of events first — that's the case most likely to break a false claim.
From Probability
The sample space is S=\{1,2,\ldots,10\}. Let A = \{1,2,3,4,5\} and B = \{4,5,6,7\}. What is \overline{A \cup B}?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
{1,2,3} is elements of A not in B — that's not the complement of the union. Check what's left in S after removing every element of A ∪ B.
{4,5} is A ∩ B — the complement of the union removes everything in EITHER set, leaving what's in neither.
This still includes 6 and 7, which belong to B (so they're already in A ∪ B) — they must be excluded from the complement.
Why
A ∪ B = {1,2,3,4,5,6,7}. Removing all of those from S = {1,...,10} leaves {8,9,10}.
First build A ∪ B by combining every element from both sets (no repeats), then find what's left over in S.
The sample space is every card in the deck. An event is any pile you sort some of those cards into. It could be "all the hearts." It could be "everything above a 7." Later in this course, it could even be "every sample whose average comes out above 70."
A sample space S is the set of every possible outcome of a random process. An event is any subset of S: a described collection of outcomes counted as "success" for some question being asked. An event doesn't have to be a single outcome. "The sample mean exceeds 70" is a perfectly good event, once "the sample" is the random process in question.
Build an event from a survey population
Ten respondents are shown. Click outcomes to build the event 'the respondent is under 30.'
Loading interactive…
Worked example
Naming an event
-
1
A single voter is picked from a survey of voters. Write the event "the voter supports the measure" as a subset statement.
-
2
Let S be every voter surveyed. The event is A = \{\text{voters who support the measure}\} \subseteq S.
-
3\boxed{A \subseteq S, \text{ the subset of "yes" voters}}
Loading interactive…
Worked example
A statistic is an event too, previewed
-
1
Later chapters will ask questions like "how likely is a sample mean above 70?" What is the sample space and what is the event, in that question?
-
2
The sample space is every possible sample the survey could have drawn, each giving its own sample mean. The event is the subset of those samples whose mean exceeds 70.
-
3\boxed{S = \text{all possible samples}, \quad A = \{\text{samples with } \bar x > 70\}}
Loading interactive…
Where this goes next
Every inference chapter ahead
Every p-value later in this course is literally P(A) for a specific event A: "a sample this extreme or more extreme than the one observed." Without a clean sense of what an event even is, that definition reads like a magic formula. It is really just the probability of a subset.
Check your understanding (counts toward your progress)
A poll surveys 500 people. Which of these is an EVENT, not the sample space?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
That's closer to the sample space itself (or the whole realized sample), not a described subset of it.
That's the sample space — every possible outcome, not a specific subset.
Why
An event is a described SUBSET of the sample space — 'answered yes' picks out some, not all, of the possible respondents.
Why does this course care about naming events precisely before computing a p-value later on?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Events apply just as well to continuous quantities like a sample mean.
Confidence intervals also rest on a precisely-named event — 'the interval captures the true parameter.'
Why
A p-value is P(an event as extreme as, or more extreme than, what was observed) — getting the event definition wrong makes the whole computation meaningless, even if the arithmetic that follows is correct.
Explain why 'the sample mean is above 70' counts as an event, even though a sample mean is a computed number, not a single outcome.
AI grading isn't configured on this deployment.
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
First mistake
Correct answer
Also accepted:
Why
Why
The sample space here is every possible sample the process could draw; each sample produces its own mean. 'Sample mean above 70' picks out the subset of possible samples whose computed mean clears 70 — a perfectly good event, just built from a computed quantity rather than a raw outcome.