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PROOF 161

Prove.

Probability lies between 0 and 1

On a probability space.

What you may use

A1P(A) ≥ 0
non-negativity (P1)
A2P(S) = 1
normalization (P2)
A3disjoint ⟹ P(A∪B)=P(A)+P(B)
additivity (P3)
A4S ∪ ∅ = S, S ∩ ∅ = ∅
the empty event — disjoint from the whole space
A5S = A ∪ Aᶜ, A ∩ Aᶜ = ∅
the complement — a disjoint partition
A6a + b = a ⟹ b = 0
cancellation of addition
A7b ≥ 0 ⟹ a ≤ a + b
addition of inequalities (order axiom)
Ea = b ⟹ apply the same operation to both sides
properties of equality

Nothing outside this list may be cited.

Lemma L1

P(∅) = 0

Proof.

Related labs

Probability lab →

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