A factory produces mineral water in bottles. When the calcium level in a bottle is less than $6.5 \mathrm { mg }$ per litre, the water in that bottle is said to be very low in calcium.

The mineral water comes from two sources, noted ``source A'' and ``source B''.
The probability that water from a bottle randomly selected from the daily production of source A is very low in calcium is 0.17. The probability that water from a bottle randomly selected from the daily production of source B is very low in calcium is 0.10.
Source A supplies $70\%$ of the total daily production of water bottles and source B supplies the rest of this production.
A water bottle is randomly selected from the total daily production. We consider the following events:\\
A: ``The water bottle comes from source A''\\
B: ``The water bottle comes from source B''\\
$S$: ``The water contained in the water bottle is very low in calcium''.

\begin{enumerate}
  \item Determine the probability of event $A \cap S$.
  \item Show that the probability of event $S$ equals 0.149.
  \item Calculate the probability that the water contained in a bottle comes from source A given that it is very low in calcium.
  \item The day after heavy rain, the factory takes a sample of 1000 bottles from source A. Among these bottles, 211 contain water that is very low in calcium. Give an interval to estimate at the $95\%$ confidence level the proportion of bottles containing water that is very low in calcium in the entire production of source A after this weather event.
\end{enumerate}