Radon contamination and Poisson statistics December 21, 2006Posted by dorigo in games, mathematics, personal, physics, science, travel.
Two weeks ago I was coming back from a trip to Chicago and I was carrying my pocket digital gamma dosimeter, something I still do when I travel on a plane. Upon walking through the corridor leading to the stairs to my apartment, I noticed that the thing beeped three times in the matter of 20 seconds or so, and the other day I decided I would investigate whether there is Radon contamination in my basement.
Radon is a inert gas emitted by the decay of Thorium, a radioactive material contained in traces in the ground. Radon is heavy and mildly radioactive itself. It accumulates over time in poorly ventilated areas such as basements, and it can stick to plastic surfaces quite easily.
My digital dosimeter records x-ray radiation and has a window of sensitivity roughly in the right region to detect decay products of radionuclides such as Radon. If set at the maximum sensitivity, it beeps every 100 nanoRems of integrated gamma dose. That means about 20+-6 seconds of exposure, under normal conditions (but beware, background radiation depends on the region of the world where you are, as well as on your latitude). By beeping three times in 20 seconds it was giving a signal, or was it a statistical fluctuation ?
I investigated the matter by going back to the basement, and taking notes of the number of beeps given in 5 minutes. The count was 22 (2.2 microRems), while outdoors it counts 15 (1.5 microRems) in the same time interval. Now the question arises, is the measurement indicating a significant contamination of Radon or just another fluke ?
If you just followed Statistics 101 and are thinking at Christmas presents with the other side of your brain, you might come out with “22 is not significantly different from 15, after all these are random counts and they are subjected to Poisson statistics, so 22 is really 22+-sqrt(22), 15 is 15+-sqrt(15), and 22+-4.6 is not so different from 15+-3.9”. Wrong.
Sure, gamma collection is a random process, but the counter does not beep every time it sees a gamma. Rather, it beeps once it collects a certain dose. For what you know, a 100 nanoRem beep could arise from one photon, or 100. In the former case you would be right in your error estimates, in the latter you would be off by a factor of 10. Further, not every photon will release the same amount of energy in your counter – but that is a unnecessary detail and I’ll neglect it here.
What we know, as I stated at the beginning, is that in normal conditions a beep occurs every 20+-6 seconds (yes, I did construct a histogram of timings to get that information). What that means is a 30% relative uncertainty on 20 seconds. That must imply that 20 seconds, or 100 nanoRems, are integrated with an equivalent number of counts which fluctuates by 30%. The number whose square root is 30% of its value is, of course, 10. So we can eyeball that 100 nanoRems are on average obtained by 10 photon counts.
Equipped with that information, we can now evaluate the significance of our 2.2 microRem/1.5 microRem difference. If 100 nanoRems are 10 counts, then on average 2.2 microRems are 220 counts. So we are really talking about 220+-15, and 150+-12. These two numbers are incompatible at almost 4-sigma level. We conclude that my basement indeed has Radon contamination, and that the radiation level in it is 1.5+-0.1 times larger than it is outdoors.
Did I know 5 minutes would have been enough to detect a significant signal before performing the experiment ? No, since I did not know how much of a signal I would be observing. But I did know that I would have been sensitive to a 20-30% increase with a 5′ exposure. Knowledge is power…