Each customer draws a raffle ticket winning a prize with probability $p \in ]0,1[$. The number $N$ of tickets distributed follows a Poisson distribution with parameter $\lambda > 0$. $X$ denotes the number of winning tickets drawn during a day, and $X$ follows a Poisson distribution with parameter $\lambda p$. We consider a real $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.1}$$ Let $V$ and $W$ be the Lambert functions defined in Part I. Using one of the functions $V$ and $W$ and Question 10, discuss according to the position of $\lambda$ with respect to $-1 - V\left(-\alpha \mathrm{e}^{-1}\right)$ the existence of a largest real $p \in ]0,1[$ satisfying condition (II.1).
Each customer draws a raffle ticket winning a prize with probability $p \in ]0,1[$. The number $N$ of tickets distributed follows a Poisson distribution with parameter $\lambda > 0$. $X$ denotes the number of winning tickets drawn during a day, and $X$ follows a Poisson distribution with parameter $\lambda p$. We consider a real $\alpha \in ]0,1[$ and the condition
$$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.1}$$
Let $V$ and $W$ be the Lambert functions defined in Part I. Using one of the functions $V$ and $W$ and Question 10, discuss according to the position of $\lambda$ with respect to $-1 - V\left(-\alpha \mathrm{e}^{-1}\right)$ the existence of a largest real $p \in ]0,1[$ satisfying condition (II.1).