Answer to Question #239199 in Statistics and Probability for ROY

Question

show that if (A/B)=1, then P(B^c/A^c)= 1

Answer 1

Definition of complement

$P(B^C|A^C)=1 -P(B|A^C)$

Bayes' rule

$P(B^C|A^C) = 1 -(\frac{P(A^C|B)P(B)}{P(A^C)})$

Definition of complement

$P(B^C|A^C) = 1 -(\frac{(1 -P(A|B))P(B)}{P(A^C)})$

Using $P(A|B) = 1$

$P(B^C|A^C) = 1 -(\frac{(0)P(A)}{P(B^C)}) \\ P(B^C|A^C) = 1$