Therefore, each toss of a needle could represent two trials. If you were able to compute the probabilities above, try this variation. Suppose we consider both sets of lines in the square grid. The first case, where L D , should be within the grasp of a beginning calculus student. You only need to be able to compute the area of the rectangle, and integrate to get the area beneath the curve.
L is a bit more involved.
Three years after publishing this analysis, I did a write-up of the solution for the second case .