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 $\square\mathbf{M}$.