Total Law of Probability Question In a community $25\%$ of the residents are smokers. Suppose $30\%$ of the smokers claim that they don’t smoke, and all non-smokers say they don’t smoke. What is the probability that when someone says that he does not smoke, he is telling the truth?
Using the total law of probability, 
$$P(A \hspace{.1cm} \text{and} \hspace{.1cm} B^c) = 0.25$$
$$P(A \hspace{.1cm} \text{and} \hspace{.1cm} B) = 0.85 + (0.3*0.25) = 0.925$$
$$P(A^c \hspace{.1cm} \text{and} \hspace{.1cm} B) = 0.85$$
from here, I'm stuck because my thought was to take the total number of people claiming they don't smoke over the number of people who don't smoke but that gives a probability greater than $1$ which is incorrect. Any tips?
 A: Just a tip.
Start with $400$ residents. 


*

*$300$ don't smoke

*$30$ do smoke and claim that they don't smoke

*$70$ do smoke and admit


How many claim that they don't smoke? And how many of them are speaking the truth on that?
A: Let $A$ be the event that the person does not smoke; let $B$ be the event that the person claims not to smoke.  Then the probability the person is telling the truth is
\begin{align*}
P(A \mid B) & = \frac{P(A \cap B)}{P(B)}\\
            & = \frac{P(B \mid A)P(A)}{P(B \mid A)P(A) + P(B \mid A^C)P(A^C)}
\end{align*}
We are given $P(A^C) = 0.25$, $P(B \mid A) = 1$, $P(B \mid A^C) = 0.3$.  Use $P(A) = 1 - P(A^C)$ to determine $P(A)$.
A: 30% of the 25% who smoke say they don't smoke. This means that 30%*25%=7.5% of the members of the community are lying, by saying that they don't smoke. 75% of the people in the community actually don't smoke. So the total percentage of people who say they don't smoke is 75%+7.5%=82.5%. As the percentage of people who actually don't smoke is 75%, the probability that somebody claiming that they don't smoke is telling the truth is 0.75/0.825, which is around 90%.
