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| Figure 1: Two of the thousands of toothpicks on my floor |
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.
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| Figure 2: Measured pi value in blue, actual in green, click for bigger version |
Imagine if you stopped at 100... probability would be broken!
ReplyDeleteCorrect me if I'm wrong, but since the event you're counting, namely that a toothpick crosses a line, is pretty likely to happen in a given trial, a Poisson distribution does not characterize the error particularly well. The variance of a binomial distribution is np(1-p), so you're missing a factor of about sqrt(.382)=.618 in your error estimate. Thus I don't think the true value of pi is quite within your error bars, but of course you would expect that to be the case in 1-.68= 32% of 1000-throw experiments.
ReplyDeleteForbes,
ReplyDeleteYou're absolutely correct. The probability of crossing was somewhere around 0.6, so it's not "rare" by a long shot. I just wanted some gauge of my error so I threw in poisson error bars as a function of habit, although, as you've pointed out, binomial would have been just as easy.
Thanks!
I was volunteering at a Pi party for middle-schoolers today, and one of the games we played was to drop toothpicks onto a big poster board scored with lines as far apart as the toothpicks were long, and then tally the ones that crossed the lines. We calculated ratios close to pi just like you did. How can I help 12 year olds understand why it works?
ReplyDeleteI don't really have a good intuitive way to come up with the expression without actually computing the probabilities. But I think it could still be properly motivated.
DeleteI have forgotten what I did and didn't know when I was 12, but I'm sure all of the kids have some sense of probability. They may not be able to compute much, but they know that when you flip a coin there's a 50-50 chance that you'll get a heads.
Given that introduction, you could probably just state the equation that gives the probability of a line crossing. A quick run-down of what each variable means should convince them that at least it is sensible. For example, using bigger toothpicks means theres a higher probability of crossing a line. That makes sense.
From here you can show that the equation could be rearrange to solve for pi. Now we have pi in terms of some things we can measure (l and d) and this thing called the probability. Here we just explain that the probability can be "measured" by counting how many times we get a crossing and dividing that by the total number of times we try. An example with the coin could drive home this point.
Even though this may not help the kids understand exactly why this works, at the very least I would think it could give them some motivation that it should work and that this number isn't just popping out of thin air.
Use circles to explain it. When you rotate the toothpick around a point it makes a circle, and that gives it an excuse to be pi-related. It's easier when you can use sin and cosine, but circles is handwavey enough for 12 year olds.
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