# A Buffoon’s Toothpicks

Figure 1: Two of the thousands of toothpicks on my floor

You’re sitting at a bar, bored out of your mind. You’ve got an unlimited
supply of pretzel rods and a lot of time to kill. The floor is made of
thin wooden planks. How can you calculate pi? This is how the problem of
Buffon’s needle was first presented to me. Stated more formally the
problem is this: given a needle of length *l* and a floor of parallel
lines separated by a distance *d*, what is the probability of a randomly
dropped needle crossing a line? Working this all out (see a derivation
here, for example), we
find that the probability a needle crosses a line is $$ P =
\frac{2l}{d\pi} $$ So now we have a nice way of experimentally coming
up with a value for pi. Simply by tossing a bunch of needles of length
*l* on a striped surface with lines separated by a distance *d* and
counting the total number of times a needle crosses a line and the total
number of throws, we can approximate the probability (and thus, pi). I
say “approximate” because it will only be true in the limit of an
infinite number of throws. Anyway, we have that $$ \frac{\mbox{Number
of crosses}}{\mbox{Number of throws}} \approx P = \frac{2l}{\pi d}
$$ so, rearranging a bit, we have that $$ \pi \approx \left(
\frac{2l}{d} \right) \left(\frac{\mbox{Number of
throws}}{\mbox{Number of crosses}} \right) $$ Now we have something
that we can go about measuring. I am going to define the following value
to be the experimental quantity we aim to measure: $$ \tilde{\pi} =
\left( \frac{2l}{d} \right) \left(\frac{\mbox{Number of
throws}}{\mbox{Number of crosses}} \right) $$ So now that we know what
we are measuring, let’s get to it! Since I’m not allowed to use needles
in my home experiments anymore, I decided to use toothpicks. For my
striped surface, I just used the wooden floor in our house (see Figure
1). The toothpick length was almost exactly the same as the distance
between lines on the floor, so we see that that the *l*and *d* terms
cancel in our expression above. To make the measurements, I threw ten
toothpicks at a time onto the floor and counted how many crossed the
lines. I chose ten because it seemed like a nice number. It was small
enough that I shouldn’t expect too much clumping of the toothpicks (and
unwanted correlations in the data), but large enough that I didn’t have
to drop and pick up a single toothpick a thousand times. I threw the
groups of ten toothpicks 100 times and tallied the results. Thus, I have
1000 throws of a single needle. It took the entirety the movie
Undercover Brother to
throw and pick up all those toothpicks, but when all was said and done I
found that out of 1000 thrown toothpicks, 618 crossed the line. Plugging
this back into our equation above (and remembering *l* = *d*), we get $$
\tilde{\pi}=\left(\frac{2l}{d}\right) \left(\frac{\mbox{Number
of throws}}{\mbox{Number of crosses}} \right)=2
\left(\frac{1000}{618}\right)=3.23$$ Well that’s not too far off I
guess. But it’s certainly not the pi that I know and love. What went
wrong? As I mentioned before, since I am only doing a finite number of
runs here I am not finding the exact probability. So is there anyway to
gauge our uncertainty? Sure. Since we are doing a counting experiment
with a lot of events, we can approximate our error using Poisson
statistics. For a Poisson distribution, the standard deviation is just
the square root of the number of events (in this case, crosses). So we
have that our total number of crosses is $$ \mbox{Number of crosses} =
618 \pm \sqrt{618} = 618 \pm 24.9 $$ So now if we want to find the
uncertainty in our final measurement, we’ll have to propagate the error
through. This gives us a final value of $$ \tilde{\pi} = 3.23 \pm
0.13 $$ and we see that the exact value of pi falls within there. We can
see that this value gets better and better if we plot our value of pi as
a function of the number of throws. Figure 2 shows the measured value of
pi (with error bars) over a wide range of throw numbers. The actual
value of pi is plotted as a green line.

Figure 2: Measured pi value in blue, actual in green, click for bigger version

So we see that the more toothpicks we drop, the closer and closer we get
to pi. Hot dog! Certainly an evening well spent.

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