\textbf{Probabilities requested will be expressed as irreducible fractions}

\textbf{Part A}\\
We flip a fair coin three times in succession. We denote $X$ the random variable that counts the number of times, out of the three flips, where the coin landed on ``Heads''.

\begin{enumerate}
  \item Specify the nature and parameters of the probability distribution followed by $X$.
  \item Copy and complete the following table giving the probability distribution of $X$
\end{enumerate}

\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$k$ & 0 & 1 & 2 & 3 \\
\hline
$P ( X = k )$ &  &  &  &  \\
\hline
\end{tabular}
\end{center}

\textbf{Part B}\\
Here are the rules of a game where the goal is to obtain three coins on ``Heads'' in one or two attempts:

\begin{itemize}
  \item We flip three fair coins:
  \item If all three coins landed on ``Heads'', the game is won;
  \item Otherwise, the coins that landed on ``Heads'' are kept and those that landed on ``Tails'' are flipped again.
  \item The game is won if we obtain three coins on ``Heads'', otherwise it is lost.
\end{itemize}

We consider the following events:

\begin{itemize}
  \item G: ``the game is won''. And for every integer $k$ between 0 and 3, the events:
  \item $A _ { k } :$ ``$k$ coins landed on ``Heads'' on the first flip''.
\end{itemize}

\begin{enumerate}
  \item Prove that $P _ { A _ { 1 } } ( G ) = \frac { 1 } { 4 }$.
  \item Copy and complete the probability tree below.
  \item Prove that the probability $p$ of winning this game is $p = \frac { 27 } { 64 }$
  \item The game has been won. What is the probability that exactly one coin landed on ``Heads'' on the first attempt?
  \item How many times must we play this game for the probability of winning at least one game to exceed 0.95?
\end{enumerate}