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 $W$.
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 $W$.