Question

If A and B are independent events and Pr(B) < 1, what is the value of Pr(A^c|B^c)?

Short answer

If A and B are independent events and Pr(B) < 1, the value \(\Pr \left( {{A^C}|{B^C}} \right)\) is \(\Pr \left( {{A^C}} \right)\).

Step-by-step solution

Given information

Events A and B are independent \(\Pr \left( B \right) < 1\) .

Computing the required probability

The two events, A as well as B, are considered to be independent if

\(P\left( {A \cap B} \right) = P\left( A \right) \cdot P\left( B \right)\)

Also,

If events A and B are independent, their complimentary events \({A^C}\) \({B^C}\)are also independent.

Thus,

\(\Pr \left( {{A^C} \cap {B^C}} \right) = \Pr \left( {{A^C}} \right) \cdot \Pr \left( {{B^C}} \right)\)

For the case,

If A and B are independent events and Pr(B) < 1, the value of\(\Pr \left( {{A^C}|{B^C}} \right)\)is obtained as:

\(\begin{aligned}{}\Pr \left( {{A^C}|{B^C}} \right) &= \frac{{\Pr \left( {{A^C} \cap {B^C}} \right)}}{{\Pr \left( {{B^C}} \right)}}\\ &= \frac{{\Pr \left( {{A^C}} \right) \cdot \Pr \left( {{B^C}} \right)}}{{\Pr \left( {{B^C}} \right)}}\\ &= \Pr \left( {{A^C}} \right)\end{aligned}\)

Therefore, the required value of \(\Pr \left( {{A^C}|{B^C}} \right)\) is \(\Pr \left( {{A^C}} \right)\).