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 $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.1}$$ We set $x = -(\lambda p + 1)$. Prove that condition (II.1) is equivalent to the condition $$x \mathrm { e } ^ { x } \leqslant - \alpha \mathrm { e } ^ { - 1 }.$$
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
$$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.1}$$
We set $x = -(\lambda p + 1)$. Prove that condition (II.1) is equivalent to the condition
$$x \mathrm { e } ^ { x } \leqslant - \alpha \mathrm { e } ^ { - 1 }.$$