\textbf{Exercise 4 — 7 points}

Theme: Probability

During the manufacture of a pair of glasses, the pair of lenses must undergo two treatments denoted T1 and T2.

\textbf{Part A}

A pair of lenses is randomly selected from production.\\
We denote by $A$ the event: ``the pair of lenses has a defect for treatment T1''.\\
We denote by $B$ the event: ``the pair of lenses has a defect for treatment T2''.\\
We denote by $\bar{A}$ and $\bar{B}$ respectively the complementary events of $A$ and $B$.\\
A study has shown that:
\begin{itemize}
  \item the probability that a pair of lenses has a defect for treatment T1, denoted $P(A)$, is equal to 0.1.
  \item the probability that a pair of lenses has a defect for treatment T2, denoted $P(B)$, is equal to 0.2.
  \item the probability that a pair of lenses has neither of the two defects is 0.75.
\end{itemize}

\begin{enumerate}
  \item Copy and complete the following table with the corresponding probabilities.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
 & $A$ & $\bar{A}$ & Total \\
\hline
$B$ & & & \\
\hline
$\bar{B}$ & & & \\
\hline
Total & & & 1 \\
\hline
\end{tabular}
\end{center}

  \item a. Determine, by justifying the answer, the probability that a pair of lenses, randomly selected from production, has a defect for at least one of the two treatments T1 or T2.\\
b. Give the probability that a pair of lenses, randomly selected from production, has two defects, one for each treatment T1 and T2.\\
c. Are the events $A$ and $B$ independent? Justify the answer.

  \item Calculate the probability that a pair of lenses, randomly selected from production, has a defect for only one of the two treatments.

  \item Calculate the probability that a pair of lenses, randomly selected from production, has a defect for treatment T2, given that this pair of lenses has a defect for treatment T1.
\end{enumerate}

\textbf{Part B}

A sample of 50 pairs of lenses is randomly selected from production. We assume that the production is large enough to assimilate this selection to a draw with replacement. We denote by $X$ the random variable which, to each sample of this type, associates the number of pairs of lenses that have the defect for treatment T1.

\begin{enumerate}
  \item Justify that the random variable $X$ follows a binomial distribution and specify the parameters of this distribution.

  \item Give the expression allowing the calculation of the probability of having, in such a sample, exactly 10 pairs of lenses that have this defect. Perform this calculation and round the result to $10^{-3}$.

  \item On average, how many pairs of lenses with this defect can be found in a sample of 50 pairs?
\end{enumerate}