\textbf{Exercise 1} (7 points) — Main topics covered: Probability

In basketball, there are two types of shots:
\begin{itemize}
  \item two-point shots: taken near the basket and score two points if successful.
  \item three-point shots: taken far from the basket and score three points if successful.
\end{itemize}
Stéphanie is practising shooting. We have the following data:
\begin{itemize}
  \item One quarter of her shots are two-point shots. Among these, $60\%$ are successful.
  \item Three quarters of her shots are three-point shots. Among these, $35\%$ are successful.
\end{itemize}

\begin{enumerate}
  \item Stéphanie takes a shot. Consider the following events:\\
  $D$: ``It is a two-point shot''.\\
  $R$: ``the shot is successful''.\\
  a. Represent the situation using a probability tree.\\
  b. Calculate the probability $p(\bar{D} \cap R)$.\\
  c. Prove that the probability that Stéphanie successfully makes a shot is equal to 0.4125.\\
  d. Stéphanie successfully makes a shot. Calculate the probability that it is a three-point shot. Round the result to the nearest hundredth.

  \item Stéphanie now takes a series of 10 three-point shots.\\
  Let $X$ be the random variable that counts the number of successful shots.\\
  Consider that the shots are independent. Recall that the probability that Stéphanie successfully makes a three-point shot is equal to 0.35.\\
  a. Justify that $X$ follows a binomial distribution. Specify its parameters.\\
  b. Calculate the expected value of $X$. Interpret the result in the context of the exercise.\\
  c. Determine the probability that Stéphanie misses 4 or more shots. Round the result to the nearest hundredth.\\
  d. Determine the probability that Stéphanie misses at most 4 shots. Round the result to the nearest hundredth.

  \item Let $n$ be a non-zero natural number.\\
  Stéphanie wishes to take a series of $n$ three-point shots.\\
  Consider that the shots are independent. Recall that the probability that she successfully makes a three-point shot is equal to 0.35.\\
  Determine the minimum value of $n$ so that the probability that Stéphanie successfully makes at least one shot among the $n$ shots is greater than or equal to 0.99.
\end{enumerate}