OCME · Bilingual mathematics & derivations

Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

Concept guide

Event: deciding what counts as an event under the full event family and under a coarser one

Roll a die once, S={1,2,3,4,5,6}. (1) In the finite elementary model F=𝒫(S), show that E={2,4,6} is an event, and decide whether E occurs when the observed outcome is 4 and when it is 3; distinguish the outcome 4 from the event {4}; describe the empty and the sure event. (2) In the general model (S,F₀) with F₀={∅,O,E,S} and O={1,3,5}: prove that F₀ is an event family (σ-algebra) on S, and prove {6}∉F₀, so although S is finite and {6}⊆S, {6} is not an event in this model.

Event

derivation-mko-event-01

Concept guide

Event independence: deciding by the product rule, with the conditional form, complements and a zero-probability event

Fair-die model: Ω={1,2,3,4,5,6}, each outcome with mass 1/6. Let A={2,4,6}, B={1,2,3,4}, D={1,3,5} and Z=∅. (1) Decide by the product definition whether A and B are independent, and with P(B)>0 check P(A|B)=P(A). (2) Prove that independence of A and B implies independence of A and Bᶜ, and check it numerically. (3) Decide whether the mutually exclusive events A and D are independent. (4) Decide whether A and the zero-probability event Z are independent, and explain why the conditional form cannot be used.

Event independence

derivation-mko-event-independence-01