Buffon’s Needle Problem

Estimate π by dropping needles on parallel lines (geometry + probability).

Setup: Parallel lines are spaced d units apart. A needle of length L is dropped at random. In this activity we will use the special case L = d (same as your handout).

Part A — Variables & Randomness

  1. The the two random variables used in the problem (see image):
    x = Vertical distance from the needle center to a parallel line going through the end of the needle
    θ = angle between needle and the lines
  2. Give the ranges (use inequalities):

Part B — When does the needle cross a line?

The needle crosses a (nearest) line when the vertical “reach” from its center to an endpoint is at least x.

  1. In the case L = d, the condition is:

    3. The probability of crossing becomes an “area ratio” in the (θ, x) plane.

    First youintegrate (find the area) from 0 to d/2sin⁡(θ):

    Then you integrate the result from θ=0 to π/2:

    Area for Crossing drops = (d/2)[0-(-1)]=(d/2)[1]=d/2

    Finally,To find the probability of crossing, you divide by the total area of the rectangle (d/2 * π/2):

Part C — Estimating π from data

Suppose you drop N needles and observe C crossings. Then P̂ = C/N and (for L = d) you can estimate:

Now you can plug in your own numbers for N and C to get an estimate for π:

For N=10000 change C to get as close as possible to π ≈ 3.14159...

Extension (optional)

In general, if L ≠ d, the probability changes. When L ≤ d, the famous formula is: P(cross) = (2L)/(πd).

Challenge: If you double the needle length, what happens to the probability? Explain in one sentence.

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