Probability Spaces: Sample Spaces, Events and Axioms
Set probability on its axioms: a sample space, events as subsets, and a measure that is nonnegative, totals one and adds over disjoint events. Every rule met earlier follows from those three.
What a learner can do afterwards
- Write the sample space and an event for an experiment
- Derive the complement and general addition rules from the axioms
- Give an assignment of probabilities that breaks the axioms and say which one
1 · Read
The dice stall gave every total a probability without ever naming what those totals were chosen from. Now name it: list every possible result of an experiment, and that list is the sample space. An event is just a collection of results from it. Toss a coin and roll a die: 2 faces times 6 faces gives 12 pairs, so that sample space holds 12 outcomes.
Three axioms govern every chance: each chance sits between 0 and 1, the whole space has chance exactly 1, and disjoint events add. The complement rule follows: P(not A) equals 1 minus P(A). Roll a die: the even event {2, 4, 6} leaves the complement {1, 3, 5}.
Add with care when events overlap: P(A or B) equals P(A) plus P(B) minus P(A and B). The overlap was counted twice, so subtract it once. With P(A) 0.5, P(B) 0.4 and P(both) 0.1, the union is 0.5 plus 0.4 minus 0.1, which is 0.8.
Check an assignment against each axiom before you trust it. A union of 1.5 breaks the 0 to 1 bound, and P(S) of 1.2 breaks the whole-space total. A Venn diagram shows why: the rectangle is the whole space and must total exactly 1.
Name the space, name the event, then let the three axioms check every chance.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.