Buffon's Needle - Using Elementary Calculus

Using Elementary Calculus

The following solution for the "short needle" case, while equivalent to the one above, has a more visual flavor, and avoids iterated integrals.

We can calculate the probability as the product of 2 probabilities:, where is the probability that the center of the needle falls close enough to a line for the needle to possibly cross it, and is the probability that the needle actually crosses the line, given that the center is within reach.

Looking at the illustration in the above section, it is apparent that the needle can cross a line if the center of the needle is within units of either side of the strip. Adding from both sides and dividing by the whole width, we obtain

Now, we assume that the center is within reach of the edge of the strip, and calculate . To simplify the calculation, we can assume that .

Let x and θ be as in the illustration in this section. Placing a needle's center at x, the needle will cross the vertical axis if it falls within a range of 2θ radians, out of π radians of possible orientations. This is the gray area in the figure. For a fixed x, we can express θ as a function of x: . Now we can let x move from 0 to 1, and integrate:

Multiplying both results, we obtain, as above.