# Find out the conditional probability

Consider I have the following probabilities:

$$P(A|B) = 0.86$$

$$P(A|B^C) = 0.35$$

$$P(B) = 0.80$$

$$P(A) = 0.758$$

Is there necessary information given to calculate $P(B^C|A^C)$? If so please guide me how. Thanks in advance.

• Is this homework? – Sven Hohenstein Apr 18 '13 at 8:25
• Nope, am revising for a test. – kype Apr 18 '13 at 9:47

The formula for the conditional probability is simply: $$P(B^c\mid A^c)=\frac{P(B^c\cap A^c)}{P(A^c)}.$$ You can calculate the denominator based on your information, so we only need to treat the numerator. Using the formula above with $B^c$ and $A^c$ interchanged, you obtain $$P(B^c\cap A^c)=P(A^c\mid B^c)P(B^c).$$ Now, try to see if you can find an expression of $P(A^c\mid B^c)$ in terms of $P(A\mid B^c)$.
• You don't need to use Bayes' theorem to find $P(A^c\mid B^c)$. Simply write out the expression and use the law of total probability. – Stefan Hansen Apr 18 '13 at 14:36
• @kype: Why not use your first formula for $P(A^c\mid B^c)$? Then $P(A^c\mid B^c)=1-P(A\mid B^c)$ which you can compute. – Stefan Hansen Apr 19 '13 at 11:19