\textbf{Candidates who have not followed the specialization course}

In preparation for an election between two candidates A and B, a polling institute collects the voting intentions of future voters.\\
Among the 1200 people who responded to the survey, $47\%$ state they want to vote for candidate A and the others for candidate B.

Given the profile of the candidates, the polling institute estimates that $10\%$ of people declaring they want to vote for candidate A are not telling the truth and actually vote for candidate B, while $20\%$ of people declaring they want to vote for candidate B are not telling the truth and actually vote for candidate A.

We randomly choose a person who responded to the survey and we denote:
\begin{itemize}
  \item A the event ``The person interviewed states they want to vote for candidate A'';
  \item $B$ the event ``The person interviewed states they want to vote for candidate B'';
  \item $V$ the event ``The person interviewed is telling the truth''.
\end{itemize}

\begin{enumerate}
  \item Construct a probability tree representing the situation.
  \item a) Calculate the probability that the person interviewed is telling the truth.\\
b) Given that the person interviewed is telling the truth, calculate the probability that they state they want to vote for candidate A.
  \item Prove that the probability that the chosen person actually votes for candidate A is 0.529.
  \item The polling institute then publishes the following results:

\begin{displayquote}
$52.9\%$ of voters* would vote for candidate A.\\
*estimate after adjustment, based on a survey of a representative sample of 1200 people.
\end{displayquote}

At the 95\% confidence level, can candidate A believe in their victory?
  \item To conduct this survey, the institute conducted a telephone survey at a rate of 10 calls per half-hour. The probability that a person contacted agrees to respond to this survey is 0.4.\\
The polling institute wishes to obtain a sample of 1200 responses.\\
What average time, expressed in hours, should the institute plan to achieve this objective?
\end{enumerate}