\textbf{Exercise 1 (7 points) -- Probabilities}

Among sore throats, one quarter requires taking antibiotics, the others do not.
In order to avoid unnecessarily prescribing antibiotics, doctors have a diagnostic test with the following characteristics:
\begin{itemize}
  \item when the sore throat requires taking antibiotics, the test is positive in $90\%$ of cases;
  \item when the sore throat does not require taking antibiotics, the test is negative in $95\%$ of cases.
\end{itemize}

The probabilities requested in the rest of the exercise will be rounded to $10^{-4}$ if necessary.

\textbf{Part 1}

A patient with a sore throat who has undergone the test is chosen at random.
Consider the following events:
\begin{itemize}
  \item $A$: ``the patient has a sore throat requiring taking antibiotics'';
  \item $T$: ``the test is positive'';
  \item $\bar{A}$ and $\bar{T}$ are respectively the complementary events of $A$ and $T$.
\end{itemize}

\begin{enumerate}
  \item Calculate $P(A \cap T)$. You may use a probability tree.
  \item Prove that $P(T) = 0.2625$.
  \item A patient with a positive test is chosen. Calculate the probability that they have a sore throat requiring taking antibiotics.
  \item a. Among the following events, determine which correspond to an incorrect test result: $A \cap T,\ \bar{A} \cap T,\ A \cap \bar{T},\ \bar{A} \cap \bar{T}$.\\
  b. Define the event $E$: ``the test gives an incorrect result''. Prove that $P(E) = 0.0625$.
\end{enumerate}

\textbf{Part 2}

A sample of $n$ patients who have been tested is selected at random. We assume that this sample selection can be treated as sampling with replacement. Let $X$ be the random variable giving the number of patients in this sample with an incorrect test result.

\begin{enumerate}
  \item Suppose that $n = 50$.\\
  a. Justify that the random variable $X$ follows a binomial distribution $\mathscr{B}(n, p)$ with parameters $n = 50$ and $p = 0.0625$.\\
  b. Calculate $P(X = 7)$.\\
  c. Calculate the probability that there is at least one patient in the sample whose test is incorrect.
  \item What is the minimum sample size needed so that $P(X \geqslant 10)$ is greater than $0.95$?
\end{enumerate}