F
= F ∩ U [where U denotes the universal set]
= F ∩ (E ∪ E') [where E' denotes the complement of E]
= (F ∩ E) ∪ (F ∩ E') [distributive property]
= E ∪ (F ∩ E') [because E is a subset of F, so F ∩ E = E]
and since these events are mutually exclusive (nothing can be in both E and E'),
P(F)
= P(E ∪ (F ∩ E'))
= P(E) + P(F ∩ E')
and therefore
P(F) - P(E) = P(F ∩ E')
and thus
P(E) ≤ P(F)
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F
= F ∩ U [where U denotes the universal set]
= F ∩ (E ∪ E') [where E' denotes the complement of E]
= (F ∩ E) ∪ (F ∩ E') [distributive property]
= E ∪ (F ∩ E') [because E is a subset of F, so F ∩ E = E]
and since these events are mutually exclusive (nothing can be in both E and E'),
P(F)
= P(E ∪ (F ∩ E'))
= P(E) + P(F ∩ E')
and therefore
P(F) - P(E) = P(F ∩ E')
and thus
P(E) ≤ P(F)