A light bulb manufacturer has two machines, denoted A and B. Machine A provides $65\%$ of production, and machine B provides the rest. Some light bulbs have a manufacturing defect:
\begin{itemize}
  \item at the output of machine $\mathrm{A}$, $8\%$ of light bulbs have a defect;
  \item at the output of machine B, $5\%$ of light bulbs have a defect.
\end{itemize}
The following events are defined:
\begin{itemize}
  \item A: ``the light bulb comes from machine A'';
  \item B: ``the light bulb comes from machine B'';
  \item $D$: ``the light bulb has a defect''.
\end{itemize}

\begin{enumerate}
  \item A light bulb is randomly selected from the total production of one day.\\
a. Construct a probability tree representing the situation.\\
b. Show that the probability of drawing a light bulb without a defect is equal to 0.9305.\\
c. The light bulb drawn has no defect. Calculate the probability that it comes from machine A.
  \item 10 light bulbs are randomly selected from the production of one day at the output of machine A. The size of the stock allows us to consider the trials as independent and to assimilate the draws to draws with replacement. Calculate the probability of obtaining at least 9 light bulbs without a defect.
\end{enumerate}