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. Define the two random variables used in the handout:
  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:
  2. The probability of crossing becomes an “area ratio” in the (θ, x) plane. Based on the handout, the result should be:

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:

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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