\begin{enumerate}
  \item Between 1998 and 2020, in France 18221965 deliveries were recorded, of which 293898 resulted in the birth of twins and 4921 resulted in the birth of at least three children.\\
a. With a precision of $0.1\%$ calculate, among all recorded deliveries, the percentage of deliveries resulting in the birth of twins over the period 1998-2020.\\
b. Verify that the percentage of deliveries that resulted in the birth of at least three children is less than $0.1\%$.
\end{enumerate}

We then consider that this percentage is negligible. We call an ordinary delivery a delivery resulting in the birth of a single child. We call a double delivery a delivery resulting in the birth of exactly two children. We consider in the rest of the exercise that a delivery is either ordinary or double. The probability of an ordinary delivery is equal to 0.984 and that of a double delivery is then equal to 0.016. The probabilities calculated in the rest will be rounded to the nearest thousandth.

2. We admit that on a given day in a maternity ward, $n$ deliveries are performed. We consider that these $n$ deliveries are independent of each other. We denote $X$ the random variable that gives the number of double deliveries performed that day.\\
a. In the case where $n = 20$, specify the probability distribution followed by the random variable $X$ and calculate the probability that exactly one double delivery is performed.\\
b. By the method of your choice that you will explain, determine the smallest value of $n$ such that $P ( X \geqslant 1 ) \geqslant 0.99$. Interpret the result in the context of the exercise.

3. In this maternity ward, among double births, it is estimated that there are $30\%$ monozygotic twins (called ``identical twins'' which are necessarily of the same sex: two boys or two girls) and therefore $70\%$ dizygotic twins (called ``fraternal twins'', which can be of different sexes: two boys, two girls or one boy and one girl). In the case of double births, we admit that, as for ordinary births, the probability of being a girl at birth is equal to 0.49 and that of being a boy at birth is equal to 0.51. In the case of a double birth of dizygotic twins, we also admit that the sex of the second newborn of the twins is independent of the sex of the first newborn. We randomly choose a double delivery performed in this maternity ward and we consider the following events:
\begin{itemize}
  \item $M$ : ``the twins are monozygotic'';
  \item $F _ { 1 }$ : ``the first newborn is a girl'';
  \item $F _ { 2 }$ : ``the second newborn is a girl''.
\end{itemize}
We will denote $P ( A )$ the probability of event $A$ and $\bar { A }$ the opposite event of $A$.\\
a. Copy and complete the probability tree.\\
b. Show that the probability that the two newborns are girls is 0.315 07.\\
c. The two newborns are twin girls. Calculate the probability that they are monozygotic.