This club makes group orders of bearings for its members from two suppliers A and B.

\begin{itemize}
  \item Supplier A offers higher prices but the bearings it sells are defect-free with a probability of 0.97.
  \item Supplier B offers more advantageous prices but its bearings are defective with a probability of 0.05.
\end{itemize}

A bearing is chosen at random from the club's stock and we consider the events:\\
$A$: ``the bearing comes from supplier A'',\\
$B$: ``the bearing comes from supplier B'',\\
$D$: ``the bearing is defective''.

\begin{enumerate}
  \item The club buys $40\%$ of its bearings from supplier A and the rest from supplier B.\\
a. Calculate the probability that the bearing comes from supplier A and is defective.\\
b. The bearing is defective. Calculate the probability that it comes from supplier B.
  \item If the club wants less than $3.5\%$ of the bearings to be defective, what minimum proportion of bearings should it order from supplier A?
\end{enumerate}