A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits. We consider $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$ With $a = \frac{p\ln(p)}{p-1}$. Let $V$ and $W$ be the Lambert functions defined in Part I. When it exists, express the largest natural integer $r$ satisfying condition (II.2) as a function of $p$, $\alpha$ and $a$ using one of the functions $V$ or $W$.

Q1 In a box, there are $n$ red balls and $(20 - n)$ white balls, where $0 < n < 20$. In each trial, a ball is taken out of the box, its color is examined, and it is returned to the box.
(1) Let $x$ be the probability that the ball taken out in one trial is red. Then, $x = \frac{n}{\mathbf{AB}}$.
(2) Let $p$ be the probability that in two trials a white ball is taken out at least once. Then $p$ can be expressed as $p = \mathbf{C} - x^{\mathbf{D}}$, where $x$ is the $x$ of (1).
(3) Let $q$ be the probability that in four trials a white ball is taken out at least twice. Then $q$ can be expressed as
$$q = \mathbf{E} - \mathbf{H}x^{\mathbf{G}} + \mathbf{H}x^{\mathbf{I}},$$
where $x$ is the $x$ of (1).
(4) For $p$ and $q$ of (2) and (3), we are to find the maximum value of $n$ such that $p < q$.
From the inequality $p < q$, we obtain the inequality
$$\mathbf{J}x^2 - \mathbf{K} \square \text{ } \square$$
When we solve this, we have
$$x < \frac{1}{\mathbf{L}}$$
Thus the maximum value of $n$ is $\square\mathbf{M}$.