The ratio of the area under the curve to the total area would be the probability of a needle crossing. We can find the area under the curve by integrating. If you do the math, you get a probability of 2/π for the case where the needle length is equal to the line spacing. There’s our pi! The reason it got in there is that the needle angle goes from negative π/2 to positive π/2 (two quarter circles). Maybe you’re not surprised after seeing that cosine function.
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