However, that change in the angle effectively changes the distance that the needle spans between the parallel lines. What math formula calculates that important span?
For the trigonometric triangle in this image, the hypoteneuse represents the needle that landed at some angle, “alpha” (‘alpha’) compared to the carpet’s lines that are parallel to the “adjacent” line AC. The length of the hypoteneuse, ‘h’, is the length of the needle.
The “opposite” line BC, has length ‘a’. That is the all-important effective length that the needle spans between the parallel lines.
By definition, “sine(alpha) = opposite/hypoteneuse”. In this picture, “sine(alpha) = a/h”. Multiply both sides by ‘h’ to solve for ‘a’. “a = h*sine(alpha)”.
Therefore, the effective length that the needle can use to span the parallel lines is “a = h*sine(alpha)”.
Materials and Setup for the Buffon Needle Drop
The materials for this math activity include:
- Note paper or a log book;
- A large sheet of paper;
- A long ruler;
- A pencil;
- Objects to toss, such as a collection of never-sharpened pencils;
- A calculator.
Prepare by writing a table to record the results. Columns should include: the test number; the number of “across”; the number of “within”; and the calculation “pi = π = ‘within‘/(‘across‘ * 2)”.
Start by drawing one straight line on the paper. Take several of the unsharpened pencils that you will toss, and lay them on the paper, perpendicular to the first line. Use those pencils to set the ruler, and draw the second line parallel to the first. Repeat a few more times, since the pencils may roll some distance.
Simplest Buffon Needle Setup: The Math
By using the length of the unsharpened pencils as “the unit of length”, and making that the “span between the parallel lines”, we’ve simplified the math.
The probability of landing across a line equals the effective length of the pencil, divided by the spanning distance between the lines. Since we set the spanning distance equal to the pencil’s length and called it “1 pencil-length unit”, we would just be dividing by one. So let’s not show that division operation.
Likewise, the length of the pencil is the hypoteneuse, ‘h’, and we define “h = 1 pencil-length unit”. Since the effective length is “a = h*sine(alpha)”, and ‘h’ has the value ‘one’, we can simply say “a = sine(alpha)”.
Why was this helpful? Remember that the probability of landing across a line is the ratio of the needle’s length divided by the spanning distance. In our diagram’s terms, the spanning distance equals the needle’s length and also equals the length of the hypoteneuse.
The probability of crossing equals “h*sine(alpha)/h”. The top ‘h’ is the length of the needle; the bottom ‘h’ is the spanning distance between the parallel lines. The upper ‘h’ cancels the lower ‘h’, so the probability of crossing is exactly “sine(alpha)”.
The value of the alpha angle (‘alpha’) may take any value from 0-180 degrees. The average value of sine(alpha) across this range equals 2/pi. (Proving this would stretch the article far beyond its editorial limits. Leave a comment if you want this proof in another article).
Finally, by running the experiment often enough, a frequentist statistician would expect the ratio of “across” to “within” results will become close to 2/pi = 0.636619772 approximately.
Although the reference articles use the above formula, it’s easier to calculate “pi = π = ‘within‘/(‘across‘ * 2)”.
Decoding Science. One article at a time.