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Chapter 3: Q. 3.26 (page 111)

show thatP(A∣B)=1, thenPBc∣Ac=1.

P(A∣B)=1, thenPBc∣Ac=1

Short Answer

Expert verified

The probability ofP(A∣B)=1⇔B⊆A⇔Ac⊆Bc⇔PBc∣Ac=1

Step by step solution

01

Step1:P(A∣B)=1

P(A∣B)=1

Logical proof: If A is certain with the knowledge of B this is equivalent to logical formulation:

B→A

There is a logical rule called modus toles that states:

B→A⇔Ac→Bc

And, in the same way that we converted probability to logic:

Ac→Bc⇒PBc∣Ac=1

02

Step2:PBc∣Ac=1

P(A∣B)=1⇔P(AB)P(B)=1⇔P(AB)=P(B)

PAcBc=PAc−PAcB

=PAc−[P(B)−P(AB)]

=PAc−0

=PAc

And going in a different direction than before yields:

PAcBc=PBc⇔PAcBcPBc=1⇔PAc∣Bc=1

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