1.3.1 · D5Probability & Statistics
Question bank — Sample spaces, events, and axioms of probability
Before we start, three words we must never blur:
- Sample space — the set of ALL outcomes, exhaustive and non-overlapping.
- Event — any subset , a bundle of outcomes we care about.
- Axioms — the three rules (non-negativity, normalization, additivity) that every valid must obey.
True or false — justify
TRUE or FALSE: Every subset of the sample space is a valid event.
For finite sample spaces, yes — any is an event; but for uncountable (like an interval) only measurable subsets qualify, which is why measure theory exists.
TRUE or FALSE: If then must be the empty set.
False — in a continuous sample space a single point can have probability yet still be a genuine outcome; "probability zero" and "impossible" only coincide in finite equally-likely spaces.
TRUE or FALSE: If then .
False for the same reason as above — can miss a probability-zero set and still have ; only in finite spaces does force to be everything.
TRUE or FALSE: Two events that are mutually exclusive cannot both have probability .
False — mutually exclusive just means ; is fine and forces , meaning together they cover the whole space.
TRUE or FALSE: The complement rule needs the outcomes to be equally likely.
False — it follows purely from Axioms 2 and 3 (since and are disjoint and union to ); it holds for any probability measure, uniform or not.
TRUE or FALSE: If then strictly.
False — monotonicity only guarantees ; the extra part can have probability , making them equal even when is a proper subset.
TRUE or FALSE: whenever and are different events.
False — being different is not enough; you need them disjoint (). Otherwise you double-count the overlap and must subtract .
TRUE or FALSE: A sample space can legally contain two outcomes that can happen on the same trial.
False — outcomes must be mutually exclusive by definition; if two things can co-occur, you've mislabelled them and the real outcome is their combination.
TRUE or FALSE: Changing what you measure changes the sample space.
True — rolling two dice gives of ordered pairs, but if you only record the sum, ; the experiment's description defines , not the physical dice.
Spot the error
"There are two outcomes for a bus arriving: on-time or late, so ." Find the flaw.
The flaw is assuming equally likely outcomes; classical needs symmetry (fair coin/die). Bus punctuality has no such symmetry — you need data or a model.
" is Kolmogorov's third axiom." Find the flaw.
Inclusion–exclusion is derived, not an axiom. Axiom 3 only covers disjoint events; the subtraction term is a theorem you prove by splitting into and .
"Since something must happen, can be any large number as long as it's the biggest." Find the flaw.
Axiom 2 pins exactly, not merely "largest." Probability is normalized to so it reads as a fraction of certainty, not an arbitrary scale.
"My model outputs , which just means very confident." Find the flaw.
Axioms 1 and 2 force ; any value above signals an algebra bug or an incoherent model, not extra confidence.
" counts everything not in , so it must be larger than ." Find the flaw.
There's no such rule; , so if then , which is smaller. Size of the complement set doesn't dictate its probability.
"Two dice: gives ." Find the flaw.
These events overlap at , so simple addition double-counts it; you must subtract per inclusion–exclusion.
" and are the same outcome, so for two dice." Find the flaw.
If dice are distinguishable, order matters and , giving outcomes. Collapsing them silently switches to a different, non-uniform sample space.
"An impossible event and a probability-zero event are always the same thing." Find the flaw.
True only in finite equally-likely spaces; in continuous spaces an outcome can be possible yet carry probability , so the concepts diverge.
Why questions
WHY must the sample space be exhaustive?
If some possible outcome were missing, the probabilities of what's left couldn't sum to (Axiom 2), and "fraction of all possibilities" would be measuring against an incomplete whole.
WHY are events defined as subsets rather than single outcomes?
Because we usually care about groups ("an even number" = ); subsets let us bundle outcomes and apply set operations (union, intersection, complement) to build the questions we actually ask.
WHY does Axiom 3 require countable additivity and not just finite?
Continuous and infinite models (like sums over all integers or limits of distributions) need the rule to hold for infinitely many disjoint events; finite additivity alone can't guarantee consistency there.
WHY can't a probability be negative?
Axiom 1 encodes that probability is a fraction of certainty; "being sure" has no meaning, so negativity would break the interpretation the whole framework rests on.
WHY do we need before we can even ask "how likely"?
"How likely" means "what fraction of what"; without defining the set of all possibilities there's no denominator, so the question is undefined — like asking "what fraction?" of nothing.
WHY does inclusion–exclusion subtract the intersection?
When you add and , every outcome in is counted twice; subtracting removes exactly one of those copies to restore the true union probability.
WHY does classical probability only work sometimes?
It secretly assumes every outcome has probability (equal likelihood); when outcomes aren't symmetric, that assumption fails and you must weight outcomes individually.
Edge cases
What is , and why?
It equals : since and are disjoint with union , Axiom 3 plus forces — the "impossible event" carries no probability.
What happens to when a coin lands on its edge, if your model said ?
The outcome falls outside your model, exposing that was not truly exhaustive; you either declare edge-landing out of scope or expand .
If and are both the certain event , what is ?
, so ; identical certain events overlap completely, and inclusion–exclusion still gives .
For a continuous , what is ?
It is — a single point has zero length/measure — even though is a perfectly possible outcome; probability lives on intervals, not points, here.
Can two events be both mutually exclusive and exhaustive?
Yes — that's exactly and : they never overlap () and together cover everything (), which is why the complement rule works.
Recall Quick self-test
One disjoint-vs-different trap ::: "Different events" is not enough for additivity; you need disjoint (). One zero-vs-impossible trap ::: In continuous spaces, probability does not mean impossible. The exact value of ::: Exactly by Axiom 2 — normalization, not merely "the largest."