Francis Comets, with whom I taught at Polytechnique in the early 2000’s, and whom I highly respected, has most sadly died on June 6. Here is a eulogy written by Patrick Cattiaux, Giambattista Giacomin, and Lorenzo Zambotti, on the site of the Société Mathématique de France (translated from the French).

Francis was a rare person. Combining both a form of conformism and a surprising originality, he marked all those who knew him: colleagues, relatives, friends. A former student of the École Normale Supérieure de Saint Cloud, in 1987 he defended a Thèse d’État at the University of Paris Sud on problems of large deviations in connection with models of statistical physics. Very quickly he demonstrated his appetite for various fields combining probability, statistics and physics. Just check out his list of publications. Francis will remain as one of the pioneers in France in the study of models in random environments for which his contributions are internationally recognized, symbolized by the Ito prize from the Bernoulli Society received in 2015. Beyond a leading scientific activity, Francis has been for forty years a major player in the structuring of mathematics in France. First assistant at the University of Paris X, then Professor at Marne la Vallée and finally at Paris 7 (which over time became Paris Diderot and now Université Paris Cité), he has, during his academic career, spared no effort in serving the community. Director of the Laboratory of Probability and Random Models, co-founder of the Fondation Sciences Mathématiques de Paris, part-time professor at the École Polytechnique, he has devoted a very large part of his time to the French mathematical community. His students and his colleagues have found in Francis listening, empathy, competence and that hint of surprise that leads to curiosity. His friends and loved ones will express the deep and personal feelings that bound them to a rare being, as mentioned above. The French mathematical community joins in the grief of his family and friends at such a painful time.

A very short book (128 pages, but with a very high price!) I received from CRC Press is Henk Tijms’ Surprises in Probability (Seventeen Short Stories).Henk Tijms is an emeritus professor of econometrics at the Vrije University in Amsterdam and he wrote these seventeen pieces either for the Dutch Statistical Society magazine or for a blog he ran for the NYt. (The video of A Night in Casablanca above is only connected to this blog through Chico mimicking the word surprise as soup+rice.)

The author mentions that the book can be useful for teachers and indeed this is a collection of surprising probability results, surprising in the sense that the numerical probabilities are not necessarily intuitive. Most illustrations involve betting of one sort or another, with only basic (combinatorial) probability distributions involved. Readers should not worry about even this basic probability background since most statements are exposed without a proof. Most examples are very classical, from the prisoner’s problem, to the Monty Hall paradox, to the birthday problem, to Benford’s distribution of digits, to gambler’s ruin, gambler’s fallacy, and the St Petersbourg paradox, to the secretary’s problem and stopping rules. The most advanced notion is the one of (finite state) Markov chains. As martingales are only mentionned in connection with pseudo-probabilist schemes for winning the lottery. For which (our very own!) Jeff Rosenthal makes an appearance, thanks to his uncovering of the Ontario Lottery scam!

“In no other branch of mathematics is it so easy for experts to blunder as in probability theory.” Martin Gardner

A few stories have entries about Bayesian statistics, with mentions made of the O.J. Simpson, Sally Clark and Lucia de Berk miscarriages of justice, although these mentions make the connection most tenuous. Simulation is also mentioned as a manner of achieving approximations to more complex probabilities. But not to the point of discussing surprises about simulation, which could have been the case with the simulation of rare events.

Ten most beautiful probability formulas (Story 10) reminded me of Ian Steward 17 formulas that changed the World. Obviously at another scale and in a much less convincing way. To wit, the Normal (or Gauss) density, Bayes’ formula, the gambler’s ruin formula, the squared-root formula (meaning standard deviation decreases as √n), Kelly’s betting formula (?), the asymptotic law of distribution of prime numbers (??), another squared-root formula for the one-dimensional random walk, the newsboy formula (?), the Pollaczek-Khintchine formula (?), and the waiting-time formula. I am not sure I would have included any of these…

All in all this is a nice if unsurprising database for illustrations and possibly exercises in elementary probability courses, although it will require some work from the instructor to link the statements to their proof. As one would expect from blog entries. But this makes for a nice reading, especially while traveling and I hope some fellow traveler will pick the book from where I left it in Mexico City airport.

This to point out an opening for a tenure track position in statistics and probability at Harvard University, with deadline December 1. More specifically, for a candidate in any field of statistics and probability as well as in any interdisciplinary areas where innovative and principled use of statistics and/or probability is of vital importance

A second Riddle(r), with a puzzle related with the integer set Ð={,12,3,…,N}, in that it summarises as

Given a random walk on Ð, starting at the middle N/2, with both end states being absorbing states, and a uniform random move left or right of the current value to the (integer) middle of the corresponding (left or right) integer interval, what is the average time to one absorbing state as a function of N?

Once the Markov transition matrix M associated with this random walk is defined, the probability of reaching an absorbing state in t steps can be derived from the successive powers of M by looking at the difference between the probabilities to be (already) absorbed at both t-1 and t steps. From which the average can be derived.

avexit <- function(N=100){
#transition matrix M for the walk
#1 and N+2 are trapping states
tranz=matrix(0,N+2,N+2)
tranz[1,1]=tranz[N+2,N+2]=1
for (i in 2:(N+1))
tranz[i,i+max(trunc((N+1-i)/2),1)]=tranz[i,i-max(trunc((i-2)/2),1)]=1/2
#probabilities of absorption
prowin=proloz=as.vector(0)
init=rep(0,N+2)
init[trunc((N+1)/2)]=1 #first position
curt=init
while(1-prowin[length(prowin)]-proloz[length(prowin)]>1e-10){
curt=curt%*%tranz
prowin=c(prowin,curt[1])
proloz=c(proloz,curt[N+2])}
#probability of new arrival in trapping state
probz=diff(prowin+proloz)
return(sum((2:length(proloz))*probz))}

leading to an almost linear connection between N and expected trapping time.