Probability · step 1 of 13

Sample spaces & events

The sample space Ω is the set of all possible outcomes of an experiment; an event is any subset of it. Probability is a function on events satisfying three axioms — non-negativity, P(Ω) = 1, and additivity for disjoint events — from which everything else follows, including P(Aᶜ) = 1 − P(A) and the addition rule.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

List the sample space explicitlyFor two dice, Ω is the 36 ordered pairs, not the 11 possible totals. Choosing an equally likely Ω is the crucial modelling decision.
Write events as subsets"The total is 7" is the set of six pairs summing to 7. Set notation makes the counting unambiguous.
Use the axioms and their consequencesP(Aᶜ) = 1 − P(A); P(A ∪ B) = P(A) + P(B) − P(A ∩ B); P(∅) = 0.
Prefer the complement when it is smaller"At least one" events are almost always easiest as 1 minus "none".

Worked example

Two fair dice are rolled. Find P(total is 7) and P(at least one six).

Ω has 36 equally likely ordered outcomes.
Total 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) — six outcomes, so 6/36 = 1/6.
At least one six: use the complement — no six on either die gives 5 × 5 = 25 outcomes.
So P(at least one six) = 1 − 25/36.

Answer. P(total 7) = 1/6 and P(at least one six) = 11/36.

Where marks get dropped

These are the specific errors that cost credit on sample spaces & events questions — QED's rubric penalises each of them separately.

Using the 11 possible totals as an equally likely sample space. They are not equally likely — 7 is six times as likely as 2.
Adding probabilities of overlapping events. P(A ∪ B) needs the intersection subtracted unless the events are disjoint.
Counting unordered pairs for two distinguishable dice, which loses the factor of 2 on mixed outcomes.

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Sample spaces & events — frequently asked questions

Why must the sample space be equally likely for counting?

Because P(A) = |A|/|Ω| assumes each outcome has probability 1/|Ω|. With unequal outcomes you must weight them individually.

Can events be independent and disjoint?

Only if one has probability 0. Disjoint events with positive probability are strongly dependent — knowing one occurred rules the other out entirely.

What is a σ-algebra?

The collection of subsets to which probability is assigned, closed under complement and countable union. For finite Ω it is usually the whole power set, so it rarely appears in exams.

The rest of Probability

Events, conditional probability, random variables. Each subtopic below has its own method, worked example and mark-losing traps.

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