Calculating probabilites

Question

We know the following probabilities:

$P(A)=0.25$

$P(A|B)=0.25$

$P(B|A)=0.5$

The question is: $P(\overline{A}|\overline{B})=?$

I have calculated: $P(B)=\frac{P(B|A)P(A)}{P(A|B)}$ and from this $P(\overline{B})=1-P(B).$

My question is, how do I get: $P(\overline{A}|\overline{B})=?$

Thank you!

Answer 1

Using the values of $P(B)$ and $P(\overline{B})$ that you have calculated, we can use the following equation to work out $P(A|\overline{B})$:- $$P(A)=P(A|B)P(B)+P(A|\overline{B})P(\overline{B})\\\Rightarrow P(A|\overline{B})=\frac{P(A)-P(A|B)P(B)}{P(\overline{B})}$$

Next we can calculate our desired solution as follows:- $$P(\overline{A}|\overline{B})=1-P(A|\overline{B})$$

Answer 2

Note that $P(A|B)=P(A)$ means that $A$ and $B$ are independent. This means that $\overline{A}$ and $\overline{B}$ are also independent, so $$ P(\overline{A}|\overline{B}) = P(\overline{A})=1-P(A) = 0.75. $$