Probably a discovery: Bad mathematics means rough scientific communication¹

G. D'Agostini

Università ``La Sapienza'' and INFN, Roma, Italia

(giulio.dagostini@roma1.infn.it, http://www.roma1.infn.it/~dagos)

Abstract:

According to the media, in spring of this year the experiment CDF at Fermilab had made most likely (``this result has a 99.7 percent chance of being correct''[1]) a great discovery (``the most significant in physics in half a century''[2]). However, since the very beginning, practically all particle physics experts did not believe that was the case. This is the last of a quite long series of fake claims based on trivial mistakes in the probabilistic reasoning that can be sketched with the following statements, understandable by everybody: the probability of a senator to be a woman is not the same as the probability of a woman to be a senator; a free neutron has only $3\times 10^{-4}$ probability to decay after two hours, but, if we observe a neutron decaying after such a time, this is not an indication of an anomalous behavior of such a particle; the fact that the probability of a Gaussian random generator with $\mu=0$ and $\sigma=1$ to produce a number, rounded to three decimal digits, equal to 3.000 is $4.2\times 10^{-6}$ does not allow us to say that, once this number has been observed, there is only $4.2\times 10^{-6}$ probability it comes from that generator, neither that $4.2\times 10^{-6}$ is the probability that 3.000 is a statistical fluctuation; and not even, still considering the latter numerical example, we can say that the probability of 3.000 to be a statistical fluctuation is $1.3\times 10^{-3}$, `because' this is the probability of such a generator to produce a number larger or equal than the observed one. The main purpose of this note is to invite everybody, but especially journalists and general public, most times innocent victims of misinformation of this kind, to mistrust claims not explicitly reported in terms of how much we should believe something, under well stated conditions and assumptions. (A last minute appendix has been added, with comments on the recent news concerning the Higgs at LHC.)