Probability Spaces: Sample Spaces, Events and Axioms
Probability has a sample space, events as subsets, and a measure governed by three axioms: it is nonnegative, totals one and adds over disjoint events. Every rule met earlier follows from these axioms.
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 naming the set those totals came from. The set of every possible result of an experiment is its sample space. An event is a collection of results from it. Flip two coins: each flip has 2 results, so the sample space {HH, HT, TH, TT} holds 2 times 2 = 4 outcomes.
Three axioms govern every chance. A chance is never negative. The whole sample space has chance exactly 1. Disjoint events, which cannot both happen, add: P(A or B) = P(A) + P(B). The complement of an event is every result outside it, and the axioms give its chance: P(not A) = 1 - P(A). Roll a die: the event 'at least 5' is {5, 6}, so its complement is {1, 2, 3, 4}.
Add with care when events overlap. Split 'A or B' into three pieces that cannot overlap: only A, both, and only B. By the third axiom, P(A or B) is those three chances added. P(A) already holds only A and both, and P(B) holds only B and both, so P(A) + P(B) counts both twice. Subtract it once: P(A or B) = P(A) + P(B) - P(A and B). With P(A) 0.7, P(B) 0.5 and P(both) 0.3, the union is 0.7 + 0.5 - 0.3 = 0.9.
Check an assignment against each axiom before you trust it. A negative chance, such as -0.1, breaks the first axiom. A whole space with chance 0.9 breaks the second. A chance cannot pass 1 either, because P(A) = 1 - P(not A) and P(not A) is never negative. So P(A) = 1.3 is not allowed.
Name the space, name the event, then let the three axioms check every chance.
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