A market gardener specializes in strawberry production.

\textbf{Part A: strawberry production}

The market gardener produces strawberries in two greenhouses denoted A and B; $55\%$ of strawberry flowers are in greenhouse A, and $45\%$ in greenhouse B. In greenhouse A, the probability that each flower produces fruit is equal to 0.88; in greenhouse B, it is equal to 0.84.

For each of the following propositions, indicate whether it is true or false by justifying the answer. An unjustified answer will not be taken into account.

\textbf{Proposition 1:}
The probability that a strawberry flower, chosen at random from this farm, produces fruit is equal to 0.862.

\textbf{Proposition 2:}
It is observed that a flower, chosen at random from this farm, produces fruit. The probability that it is located in greenhouse A, rounded to the nearest thousandth, is equal to 0.439.

\textbf{Part B: strawberry packaging}

Strawberries are packaged in trays. The mass (expressed in grams) of a tray can be modeled by a random variable $X$ which follows the normal distribution with mean $\mu = 250$ and standard deviation $\sigma$.

\begin{enumerate}
  \item We are given $P ( X \leqslant 237 ) = 0.14$. Calculate the probability of the event ``the mass of the tray is between 237 and 263 grams''.
  \item Let $Y$ be the random variable defined by: $Y = \frac { X - 250 } { \sigma }$.\\
a. What is the distribution of the random variable $Y$?\\
b. Prove that $P \left( Y \leqslant - \frac { 13 } { \sigma } \right) = 0.14$.\\
c. Deduce the value of $\sigma$ rounded to the nearest integer.
  \item In this question, we assume that $\sigma$ equals 12. We denote by $n$ and $m$ two integers.\\
a. A tray is compliant if its mass, expressed in grams, lies in the interval $[ 250 - n ; 250 + n ]$. Determine the smallest value of $n$ for a tray to be compliant with a probability greater than or equal to $95\%$.\\
b. In this question, we consider that a tray is compliant if its mass, expressed in grams, lies in the interval $[230; m ]$. Determine the smallest value of $m$ for a tray to be compliant with a probability greater than or equal to $95\%$.
\end{enumerate}