Probability to find A all. I think I need some help:
Suppose the answer of all questions were A.
If you are smart, the probability to find A is 0.001.
If you are not smart, the probability is only 0.00001.
And the probability to be smart is 0.3.  


*

*What is the probability to find A?

*What is the probability that you are smart
a) if you find A?
b) if you not find A?


In the first task is it just 0.001*0.00001=0.00000001?
Thank you!
 A: Maybe draw a tree diagram. There are two branches, Smart and Not Smart. The branch Smart has $0.3$ written beside it, the branch Not Smart has 0.7$ written beside it.
From Smart, there are two branches, Find A and Not Find A. The Find A branch has $0.001$ written beside it. There are similarly two branches leading from Not Smart.
The probability of finding A is the sum of the probabilities over all paths that lead to Find A. The required probability is $(0.3)(0.001)+(0.7)(0.00001)$.
We can also derive the answer using conditional probability notation. Write $S$ for Smart, $S'$ for Not Smart, $A$ for Find A.
We want $\Pr(A)$. This is $\Pr(A|S)\Pr(S)+\Pr(A|S')\Pr(S')$. All the required components are known.
A: Well, the probability of being smart is $0.3$ and the probability of finding $A$ if you're smart is $0.001,$ so the probability of being smart and finding $A$ is $0.3\cdot0.001=0.0003.$ Now, what is the probability of not being smart? Hence, what is the probability of finding $A$ and not being smart? Hence, what is the probability of finding $A$?
For part $2$, we will use the answers to part $1$. If the probability of finding $A$ is $p,$ then the probability of being smart if you find $A$ is $\frac{0.0003}p.$ Now, the probability of being smart and not finding $A$ is $0.001-0.0003=0.0007,$ and the probability of not finding $A$ is $1-p.$ Hence, the probability of being smart if you don't find $A$ is $\frac{0.0007}{1-p}.$
