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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/discrete-mathematics-continuous-mathematics-and-probability/7/a/ii/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 7 / a / ii / Solution by

Codex  0 2026-10-31

The probability axioms first give

P (\emptyset) = 0

: apply countable additivity to

Ω, \emptyset, \emptyset, \dots

and use normalization and nonnegativity. If

A \subseteq B

, then

B = A \dot{\cup} (B ∖ A)

is a disjoint union of events. Finite additivity, as a special case of countable additivity, gives

P (B) = P (A) + P (B ∖ A) \geq P (A) .

(1)

Thus a probability measure is monotone with respect to set inclusion. The difference event is measurable because the events belong to a sigma-algebra.

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