<s>[INST] Probability Interview Question - Brain teaser 
Player A has a thirty-sided and Player B has a twenty-sided die. They both roll the dies and whoever gets the higher roll wins. If they roll the same amount Player B wins. What is the probability that Player B win?

So I have
$$1 - \left(\frac{1}{3}+\frac{1}{2}\cdot\frac{2}{3}-\frac{1}{30}\right) = \frac{11}{30}.$$
There is $100\%$ chance of winning when A roll from 20 - 30 but for the rest $2/3$ there is only $50\%$ chance of winning, in addition, there is $1/30$ of chance that player A could roll the same number as B
The answer for this question should be $0.35$, I am not sure where I did wrong. 
I just figured maybe I should do this instead 
$1 - (\frac{1}{3}+(\frac{1}{2}-\frac{1}{30})\cdot\frac{2}{3}) = \frac{16}{45}$ is this right ?
 [/INST] Let $A$ throw $1..30$.
$B$ has no chance against $21..30$ with probability $\frac{1}{3}$: it wins $0\times\frac{1}{3}$ such throws.
$B$ wins $\frac{2}{3}\times\frac{1}{20}$ equal throws.
$B$ also wins half of the remaining $19$ throws of $A$: $\frac{1}{2}\times\frac{19}{20}$.
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