Buffon's needle

Let the probability of the needle touch at least one of the brown lines at direction $\theta$ is $P(\theta)$ .

because only the ratio of $L$ to $d$ make sense to this question but not the value of $L$ and $d$ separately, let the ratio $\lambda = \frac{d}{L}$ , for this question, $\lambda=0.2$ .

When $sin\theta \leq \lambda$ ,

$P(\theta)= \frac{sin\theta}{\lambda}$

When $sin\theta > \lambda$ ,

$P(\theta)=1$ , this says that when $\theta$ between some interval, the needle must touch at least one of the brown lines.

Using Desmos Graphing Calculator,

The shaded region is the area under $P(\theta)$ .

Because all direction are equally distributed,

$A = \frac{Area_{shadedregion}}{\pi}$ , denominator is the length of horizontal.

Noted that orange line represent $x=sin^{-1} \lambda$ , purple line represent $x=\pi - sin^{-1} \lambda$ .

$\pi A = Area_{shadedregion} =2\int_0^{sin^{-1}\lambda} \frac{sin\theta}{\lambda}d\theta +\pi - 2sin^{-1} \lambda$

After some calculation,

$A = \frac{2}{\lambda \pi}(1-\sqrt{1-\lambda^2})+1-\frac{2}{\pi}sin^{-1}\lambda = 0.9361232232$

So,

$\lfloor 10000A \rfloor = \boxed{9361}$ .