Numberphile has a video about the Buffon's needle experiment (Video). I am writing an essay on determining $\pi$ using probability and I need to show my understanding of the topic. I kind of already understand how we get $\pi$ from the randomness however I need some clarification on how the following was evaluated:
$$\int_{\theta=0}^{{\pi\over2}}\int_{x=0}^{{l\over2}sin\theta}P_x P_{\theta} dx d\theta$$
$P_x$ being ${1\over l}$ where $l$ is the length of the match (this should be irrelevant since it will cross out)
$P_{\theta}$ being ${2 \over \pi}$
The result should be ${1\over\pi}$
I would recommend watching the video, start at 3 min if you only want the math behind it.
I need someone to explain a step by step of how we got to the result, specifically how double integration works in this case. I do understand some calculus so no need to go to extreme details.