Question

a needle of length $l$ is dropped onto a floor made of parallel boards of width $d$. make the obvious uniformity and independence assumotions concerning the position and orientation of the fallen needle. show that if $l < d$, then the probability that the needle intersects a crack between the two floorboards is $2l/\pi d.$

Answer 1

The question involves Buffon's needle problem in probability theory, where the probability that a needle intersects a crack between floorboards is found to be 2l/πd under the assumption that the length of the needle is less than the width of the boards.

Step-by-step explanation:

The question concerns a classic problem in probability theory known as Buffon's needle problem, which is used to estimate the value of π and involves geometric probabilities. Calculating the probability that a needle of length l intersects a crack between two floorboards, where the boards have a width of d and assuming l < d, involves integrating over the continuous uniform distribution of the needle's position and orientation.

Without loss of generality, we can consider the position of the needle's midpoint and its angle relative to the cracks, as these are the two independent factors that determine whether the needle crosses a crack or not.

Let the angle between the needle and the cracks be θ, uniformly distributed between 0 and π, and let the distance of the needle's midpoint from the nearest crack be x, uniformly distributed between 0 and d/2. The needle crosses a crack if x < (l/2) × sin(θ). The probability of this is the ratio of the area under the curve x = (l/2) × sin(θ) over the rectangle formed by the 0 to d/2 and 0 to π intervals.

The resulting probability, after integrating and simplifying, yields the well-known result P = 2l/πd.