Berwin, thanks for your response.

It is not clear what are the grains of rice and what are the tiles.  I'd assumed that the grains of rice are 100 staff, and that the probability of falling on one particular tile is the probability of 1 in  400 that any individual staff member will be sick.   If as you suggest the idea was that 100 staff with the disease are spread across 400 labs, I'd think that the disease is not all that rare.  And on your calculation, the probability of 0.0025 for five in one of the 400 labs still strongly suggests that the occurrences are not independent, and makes nonsense of his claim that:

"We, know-­all professors, had fallen into a gross statistical error. We had become convinced that the “above average” number of sick people required an explanation."

Such an occurrence, in one of 400 labs, does require explanation. There is an issue -- does the individual lab look at the probability that it will have five cases, or at the probability that one of 400 somewhere in the country will have 5 cases (and in the latter case, one should be checking across all 400 labs)?

An additional point is that the 100 occurrences across 400 labs (if that is what Rovelli had in mind) should be compared with the disease incidence in the general population.  Is that number the expected number, given its incidence in the population.

As a side comment, if experiments treated their p < 0.05 as just what one might expect when 20 labs are doing a broadly comparable experiment, there'd be many fewer papers published!  As you suggest, it is the old chestnut, "What is the relevant reference population?"

Cheers,
John.

Berwin Turlach wrote:

Hi John,

Thank you for the link to an interesting article.

Well, yes, if one is in a dining room with 400 tiles and throws 100 rice grains into the air, the rice grains will probably not fall uniformly onto the tiles.  I would probably choose a bivariate normal distribution (or some other elliptic distribution) as a model (but how to choose all the parameters for that distribution?).

But even if we assume that every rice grain is equally likely to fall on any of the tiles and that the rice grains fall independently of each other,  your calculation addresses the probability that a particular tile will have x rice grains on it.  Whereas the point seems to be to look at the probability that there is at least one tile with x rice grains on it.  Pretty much as in the Birthday problem.  The difference of whether we ask for a match for one particular day, or a match for any day.

So I think a relevant simulation would be:

> throw <- function(x){ tiles <- sample(1:400, 100, replace=TRUE) ; any(table(tiles)==x)}
> mean(replicate(10000, throw(2)))
[1] 0.9999
> mean(replicate(10000, throw(3)))
[1] 0.5567
> mean(replicate(10000, throw(4)))
[1] 0.0472
> mean(replicate(10000, throw(5)))
[1] 0.0025

So the chance of finding at least one tile with 3 rice grain on it is ~ 0.55.  The chance to have a tile with 4 or 5 rice grains on it are small, but not as small as in your calculation.

I guess the mathematician thought of the rice grains as "persons who have a disease" and the tiles as "possible labs".  And he wanted to illustrate that the probability of having a lab with a concentration of cases is not as small as people might think it is (if there are many other similar labs around the world).  So is there something going on in this particular lab where one has observed the concentration of cases?  Or is it just the unlucky one that we would expect to see among all the labs that exists?  How to set up the probability calculation post-hoc after having observed the cluster of cases? 

Other examples that come to mind are discussions of the effect on one's health when living close to nuclear power plants, close to overland power lines, wind turbines, WiFi modems, and so forth.  Some of these discussions are, of course, more scientific and serious than others. :-)

Cheers,

Berwin

A recent Guardian article has the title "Statistical illiteracy isn't a niche problem. During a pandemic, it can be fatal"
https://www.theguardian.com/commentisfree/2020/oct/26/statistical-illiteracy-pandemic-numbers-interpret
It starts thus

<<<
In the institute where I used to work a few years ago, a rare non-infectious illness hit five colleagues in quick succession. There was a sense of alarm, and a hunt for the cause of the problem. In the past the building had been used as a biology lab, so we thought that there might be some sort of chemical contamination, but nothing was found. The level of apprehension grew. Some looked for work elsewhere.

One evening, at a dinner party, I mentioned these events to a friend who is a mathematician, and he burst out laughing. “There are 400 tiles on the floor of this room; if I throw 100 grains of rice into the air, will I find,” he asked us, “five grains on any one tile?” We replied in the negative: there was only one grain for every four tiles: not enough to have five on a single tile.

We were wrong. We tried numerous times, actually throwing the rice, and there was always a tile with two, three, four, even five or more grains on it.

. . . We, know-­all professors, had fallen into a gross statistical error. We had become convinced that the “above average” number of sick people required an explanation. Some had even gone elsewhere, changing jobs for no good reason.
>>>

Assuming rice grains fall independently on tiles, the probabilities of 0, 1, . , , 5, 6+ grains on one tile are, with the R code used:

setNames(round(dbinom(0:5,100,.0025), 5), 0:5)
      0       1       2       3       4       5
0.77856 0.19513 0.02421 0.00198 0.00012 0.00001 

Thus, the probability of 5 grains on the one tile is vanishingly small. (The author does not give this calculation.)

Then, what is the author's point with this example?  That rice grains that are thrown randomly by a bunch of mathematicians are unlikely to fall independently.  If so, what is the relevance to the story of the five colleagues who had a rare infectious illness (probability for one such illness, maybe 1 in 400, or 0.0025)?  If so, then, surely the point is that the five illnesses are very unlikely to be independent events, and that is makes sense to look for a common cause.   It appears to me that the author is himself guilty of a "gross statistical(?) error".  Have I missed something.

For anyone who wants to simulate the probabilities, the following code may be used:

n <- numeric(7)
for(i in 1:100000){
  sam <- sample(1:400,100, replace=T)
  tab <- table(sam)
  for(j in 2:6)n[j]<-n[j]+sum(tab==j-1)
  n[1]<-n[1]+400-length(tab)
  n[7] <- n[7]+sum(tab>5)
}
setNames(round(n/sum(n),5), c(0:5,"6+"))
      0       1       2       3       4       5      6+
0.77854 0.19516 0.02421 0.00197 0.00012 0.00001 0.00000