I'm trying to prove the following problem:
Show that $RP$ is closed under concatenation
Now, let's say that the two languages are $L_{1}$ and $L_{2}$ (both in $RP$). Then I accept a word iff the TM of $L_{1}$ accepted the first part of the word (with a probability $\geq\frac{1}{2}$), and the TM of $L_{2}$ accepted the second part of the word (with a probability $\geq\frac{1}{2}$).
But then each word in the language $L_{1} \circ L_{2}$ would be accept with a probability of $\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}$, so the language $L_{1} \circ L_{2}$ will not be in $RP$.
So what is the correct proof to the problem?