A store is equipped with self-service automatic checkouts where the customer scans their own items. The checkout software regularly triggers verification requests.

The check can be either ``complete'': the store employee then scans all of the customer's items again; or ``partial'': the employee then selects one or more of the customer's items to verify that they have been scanned correctly.

If a check is triggered, it is a complete check one time out of ten.\\
When a complete check is triggered, a customer error is detected in $30\%$ of cases.\\
When a partial check is performed, in $85\%$ of cases, there is no error.\\
A check is triggered at an automatic checkout.\\
We consider the following events:
\begin{itemize}
  \item T: ``The check is a complete check'';
  \item E: ``An error is detected during the check''.
\end{itemize}
We denote $\bar{T}$ and $\bar{E}$ the complementary events of $T$ and $E$.

\begin{enumerate}
  \item Construct a probability tree representing the situation and then determine $P(\bar{T} \cap E)$.
  \item Calculate the probability that an error is detected during a check.
  \item Determine the probability that a complete check was performed, given that an error was detected. The answer will be given rounded to the nearest hundredth.
\end{enumerate}