The magic of the d’Alembert

Simulations of the d’Alembert on a faIr roulette wheel with 36 paying outcomes and one “0”. Even odds bets (e.g., red versus black). Each line is one game. Each picture is 200 games. Parameters: initial capital of 25 units, maximum number of rounds is 21, emergency stop if capital falls below 15.


Harry Crane and Glenn Shafer (2020), Risk is random: The magic of the d’Alembert.

Stewart N. Ethier (2010), The Doctrine of Chances Probabilistic Aspects of Gambling. Springer-Verlag: Berlin, Heidelberg.

startKapitaal <- 25
eersteInzet <- 1
noodstopKapitaal <- 15
aantalBeurten <- 21
K <- 100
J <- 200
winsten <- rep(0, K)

for (k in (1:K)){

	plot(x = -2, y = -1, ylim = c(-5, 45), xlim = c(0, 22), xlab = "Beurt", ylab = "Kapitaal")
	abline(h = 25)
	abline(h = 0, col = "red")

	aantalKeerWinst <- 0
	totaleWinst <- 0

	for (j in (1:J)) {

		huidigeKapitaal <- startKapitaal
		huidigeInzet <- eersteInzet
		resultaten <- sample(x = c(-1, +1), prob = c(19, 18), size = aantalBeurten, replace = TRUE)
		verloop <- rep(0, aantalBeurten)
		stappen <- rep(0, aantalBeurten)
		for(i in 1:aantalBeurten) {
			 huidigeResultaat <- resultaten[i]
			 if(huidigeInzet > 0){
				  stap <- huidigeResultaat * huidigeInzet
				  stappen [i] <- stap
				  huidigeKapitaal <- huidigeKapitaal + stap
				  huidigeInzet <- max(1, huidigeInzet - stap)
				  if(huidigeKapitaal < noodstopKapitaal) {huidigeInzet <- 0}
				  verloop[i] <- huidigeKapitaal
			 } else {
				  stappen[i] <- 0
				  verloop[i] <- huidigeKapitaal
	aantalKeerWinst <- aantalKeerWinst + (verloop[aantalBeurten] > startKapitaal)
	totaleWinst <- totaleWinst + (huidigeKapitaal - startKapitaal)
	lines(0:aantalBeurten, c(startKapitaal, verloop) + runif(1, -0.15, +0.15 ), add = TRUE)
print(c(k, aantalKeerWinst, totaleWinst))
winsten[k] <- totaleWinst

The program repeatedly runs and plots 200 games of each maximally 21 rounds. Below are the total number of times that the player made a profit, and the final net gain, for 100 sets of 200 games. The sets are numbered 1 to 100.

[1]    1  100 -483
[1]    2  108 -336
[1]    3  103 -517
[1]    4  110 -275
[1]   5 123 -40
[1]   6 125 148
[1]    7  115 -209
[1]    8  104 -427
[1]    9  108 -356
[1]   10  110 -225
[1]   11  101 -440
[1]  12 120  80
[1]   13  108 -334
[1]   14  110 -279
[1]   15   99 -538
[1]   16  114 -101
[1]  17 113 -92
[1]  18 117 -87
[1]   19  104 -363
[1]   20  103 -320
[1]  21 114 -52
[1]   22  107 -422
[1]   23  108 -226
[1]   24  115 -173
[1]   25  110 -209
[1]   26  109 -261
[1]   27  114 -186
[1]  28 120 -62
[1]  29 123  35
[1]   30  101 -442
[1]   31  111 -215
[1]   32  104 -378
[1]  33 120  49
[1]  34 117 -49
[1]   35  119 -102
[1]   36  104 -488
[1]   37  107 -402
[1]  38 122  38
[1]   39  100 -549
[1]  40 116 -31
[1]  41 127 220
[1]   42  105 -427
[1]   43  114 -153
[1]   44  109 -256
[1]   45  119 -166
[1]  46 121  47
[1]   47  105 -417
[1]   48  113 -134
[1]  49 121 111
[1]   50  112 -307
[1]  51 114 -92
[1]  52 123 123
[1]  53 118  24
[1]   54  113 -188
[1]  55 124 127
[1]   56  110 -229
[1]   57  113 -255
[1]   58  101 -554
[1]   59  114 -345
[1]  60 124 236
[1]   61   97 -599
[1]   62  115 -220
[1]  63 120  55
[1]   64  102 -512
[1]  65 121 109
[1]   66  112 -219
[1]   67  112 -181
[1]  68 115 -45
[1]   69  107 -474
[1]   70  109 -272
[1]   71  116 -134
[1]   72  107 -440
[1]   73  108 -470
[1]  74 119 -85
[1]  75 115   1
[1]  76 115 -88
[1]   77  113 -219
[1]  78 118 -55
[1]   79  115 -150
[1]  80 124  70
[1]   81  115 -203
[1]   82  115 -153
[1]   83  109 -219
[1]   84   97 -675
[1]   85  108 -396
[1]   86  112 -220
[1]   87  115 -187
[1]   88  108 -290
[1]   89  114 -182
[1]   90  105 -439
[1]   91  113 -183
[1]   92  115 -216
[1]  93 124 110
[1]   94  115 -173
[1]  95 125 177
[1]   96  110 -203
[1]  97 128 160
[1]  98 114 -83
[1]  99 118 -90
[1] 100 123 106

A fungal year

I want to document the more than 20 species of wild mushrooms which I’ve collected and enjoyed eating this year. I will go through my collection of photographs in reverse chronological order. But above, the featured image, taken back in September: Neoboletus luridiformis, the scarletina bolete; in Dutch, heksenboleet (witch’s bolete. Don’t worry. The guy to avoid is the devil’s bolete).

I get my mushroom knowledge from quite a few books and from many websites. In this blog I will just give the English and Dutch wikipedia pages for each species. I highly recommend Google searching the Latin name (though notice – scientific names do change, as science gives us new knowledge) and if your French, German or other favourite language also has a wikipedia page, nature lover’s web pages, forager’s webpages, or whatever, check them out, because ideas of edibility and of how to cook mushrooms which are considered edible varies all over the world. If at some time there was a famine, and the only country people who could survive were those who went out in the forest and found something they could eat, then their fellows who had allergic reactions to those same mushrooms did not survive, and in this way different human populations are adapted to different fungi populations. It’s also very important to consult local knowledge (in the form of local handbooks, local websites) since the dangerous poisonous look-alikes which you must avoid vary in different parts of the world.

Do not eat wild mushrooms raw. You don’t know what is still crawling about in it, and you don’t know what has pooped or pissed on it or munched at it recently. Twenty minutes gentle cooking should destroy anything nasty, and moreover, it breaks down substances which are hard for humans to digest. The rigid structure of mushrooms is made of chitin (which insects use for their external body) and we cannot digest it raw. Some people have allergic reactions to raw chitin.


Paralepista flaccida

Russula cyanoxantha

Armillaria mellea

Coprinus comatus

Suillus luteus

Amanita muscaria

Sparassis crispa

[To be continued]

Appendix: some mushrooms and fungi to be wondered at, but not eaten

1. Paralepista flaccida

Tawny funnel, Roodbruine schijnridderzwam. Grows in my back garden in an unobtrusive spot, fruiting every year in December to January. Yellow-pinkish spore print, lovely smell, nice taste. Also after frying! The combination of aroma/taste/spore-print just does not fit any of the descriptions of this mushroom or those easy to confuse with it which I can find. There is a poisonous lookalike which however is not supposed to taste good, so that’s why I dared to eat this one. It grows close to a Lawson cyprus but there may be other old wood remains underground in the same spot.

English wikipedia:

Netherlands wikipedia:

2. Russula cyanoxantha

Charcoal burner, Regenboogrussula (rainbow russula). Very common in the forests behind “Palace het Loo”. A really delicious russula species, easy to identify.

English wikipedia:

Netherlands wikipedia:

3. Armillaria mellea

Honey fungus, echte honingzwam. These fellows are growing out of the base of majestic beech trees at Palace het Loo. The trees are all being cut down now; excuse: “they’re sick”; true reason: high quality beech wood is very valuable. The trees are hosts to numerous fungi, animals, birds. The managers of the park have been doing their best to kill them off for several decades by blowing their fallen leaves away and driving heavy machinery around. Looks like their evil designs are bearing fruit now.

English wikipedia:

Netherlands wikipedia:

4. Coprinus comatus

Shaggy ink cap, Geschubde inktzwam. One of the last ones of the season, very fresh, from a field at the entrance to the Palace park. These guys are so delicious, fried in butter with perhaps lemon juice, and a little salt and pepper, they have a gentle mushroom flavour, they somehow remind me of oysters. And of Autumns in Aarhus, picking them often from the lawns of the university campus.

English wikipedia:

Dutch wikipedia:

5. Suillus luteus

Suillus luteus

Slippery jack, bruine ringboleet. This one looks rather slimy and it is said that it needs to be cooked well, it disagrees with some people. It didn’t disagree with me at all, but I must say it did not have much flavour, and does feel a bit slippery in your mouth.

English wikipedia:

Dutch wikipedia:

6. Amanita muscaria

Fly agaric, vliegenzwam. This mushroom contains both poisons and psychoactive substances. However, both are water soluble. One therefore boils these mushrooms lightly for 20 minutes in plenty of lightly salted water with a dash of vinegar, then drain and discard the fluid; then they can be fried in butter and brought up to taste with salt and pepper. They are then actually very tasty, in my opinion.

Another use for them is to soak them in a bowl of water and leave in your kitchen. Flies will come and investigate it, taste some get high (literally and figuratively) and drop dead. The smell is pretty disgusting at this stage.

I understand you can dry them, grind to powder, and make tea. This allegedly destroys the poisons but leaves enough of the psychoactive substances to have interesting effects. I haven’t tried it, since one of the effects is to set your heart racing, and since I have a dangerously irregular hearth rhythm already, I should not experiment with this.

Some people munch a small piece raw, from time to time, while walking in the forests. I have tried that – teaspoon size, desertspoon size even, without noticing anything except that perhaps for a moment everything sparkled more beautifully than usual. Probably that was the placebo effect.

Amanita muscaria is not terribly poisonous. If you cook and eat three or four you will probably throw up after an hour or two and also experience rather unpleasant hallucinations. To be rounded off with diarrhea and generally feeling unwell. You might find yourself getting very large or very small, it depends of course whether you nibble from the right-hand edge of the mushroom or the left-hand edge. You might believe you can fly so it can be dangerous to be in high places on your own. The poisons may damage your liver but being water soluble they are quite efficiently and rapidly excreted from the body, which is a good thing, so eating them just once probably won’t kill you and probably won’t give you permanent damage. Several other Amanita species are deadly poisonous. With poisons which do not dissolve in water and do not leave your body after you’ve eaten them, but instead destroy your liver in a few days. One must learn to recognise those mushrooms very well. In my part of the world: Amanita phalloides – the death cap (groene knolamaniet); Amanita pantherina – the panther cap (panteramaniet). I have seen these two even in the parks and roadside verges in my town, as well as in the forests outside. More rare is Amanita virosa – the destroying angel (kleverige knolamaniet). But I believe I have seen it close to home, too. It is a white mushroom with white gills and consequently many people believe you must never touch a white mushroom with white gills. Consequently, writers of mushroom books themselves generally have the idea that edible white mushroom with white gills, which do exist, do not taste particularly good, either, and so one should not bother with them. Hence they do not explain well how you can tell the difference. We will later (i.e., earlier this year) meet the counterexample to that myth.

Because of the psychoactive effects of Amanita muscaria it is actually presently illegal, in the Netherlands, to be found in possession of more than a very small amount.

7. Sparassis crispa

The cauliflower mushroom, grote sponszwam. One of my favourites. It does have the tendency to envelope leaves and insects in its folds. Before cooking it has a wonderful aroma, almost aromatic, but on frying it seems to lose a lot of flavour.

English wikipedia:

Dutch wikipedia:

RIP Bill van Zwet

The photograph above was taken by me at my summerhouse (i.e., an allotment garden with a comfortable large shed) in Leiden, exactly 9 years ago, with Jerry Friedman, Jacqueline Meulman, and Willem. It is early evening and we are enjoying a choice single malt and some tasty snacks. Jacqueline is wearing a t-shirt with the logo designed by me of our new master programme “Statistical Science for the Life and Behavioural Sciences”.

Below I am, for the time being, just posting a large collection of photographs sent in by a number of Bill’s old friends. I will also later add some of the comments they made in their emails. I will perhaps also add some personal remarks in the near future.

Before the many photo albums contributed by Bill’s friends, here is a link to a Zoom commemoration hosted by myself, which started one hour after the start of Willem’s funeral. Participants: Maryse Loranger, Richard Gill, Estate Khmaladze; Friedrich Götze and his wife, Marie-Colette van Lieshout, Nick Fisher; Peter Grunwald, and Ildar Ibragimov (who later managed to switch on his webcam); later arrivals were Ronald Cramer and Steffen Lauritzen:

The file is 207 MB mp4; duration 35 minutes; 5 minutes silence at ca. 25 – 30 min., when Zoom briefly fails us. You can also watch it on YouTube:

And now to the photographs. First of all, three sets of pictures taken by Chris Klaassen, starting with Willem’s 75th birthday celebrations:

Chris Klaassen, Willems 80th birthday celebration, Leiden:

Chris Klaassen, Willem’s 80th birthday event in Utrecht:

David Mason:

Friedrich Götze:

Jacqueline Meulman and Maarten Kampert:

Marie Huskova:

Marta Fiocco:

Nick Fisher:

Niels Keiding:

© Niels Keiding

Peter Bickel:

Richard Gill:

Rudi Beran:

Sara van de Geer:

Stephen Stigler:

Vera Wellner:


The Beginning of the End, or the End of the Beginning?

Fhloston Paradise interior film frame

We see the hotel lobby of the Fhloston Paradise hotel, the enormous space cruise-ship from Luc Besson’s movie “The Fifth Element”. It occurs to me that our global village, the Earth, has itself become a huge space cruise-ship, including the below-decks squalor of the quarters of the millions of people working away to provide the luxury for the passengers in the luxurious areas in the top-decks.

Now turn to some other pictures. Covid-19 bar-charts.

No photo description available.

From top to bottom: (per day) new proven infections, new hospital admissions, deaths, in the Netherlands. Source: Arnout Jaspers. It looked to Arnout that we were already past the peak of the epidemic. His source: RIVM,

The curves look to me like shifted and shrunk versions of one another. About a third of those who are reported infected (mostly because they actually reported themselves sick) get so bad they go to hospital a small week later and a quarter of them die there just a few days later.

People who are infected (and infectious) but don’t realise it are not in these pictures. There have been an awful lot of them, it seems. Self-isolation is reducing that number.
As Arnout figured out for himself by drawing graphs like this, and David Spiegelhalter reported earlier in the UK, this pandemic is in some sense (at present) not really such a big deal. Essentially, it is doubling everyone’s annual risk of death this year and hopefully this year only. This means that 2% of all of us will die this year instead of the usual 1%. It looks as though the factor (two) is much the same for different age-groups and different prior health status. The reason this has such a major effect on society is because of “just-in-time” economics which means that our health care system is pretty efficient when the rate is 1% but more or less breaks down when it is 2%.

What is alarming are reports that younger people are now starting to get sicker and die faster than originally was the case. Human-kind is one huge petri-dish in which these micro-machines [“The genome size of coronaviruses ranges from approximately 27 to 34 kilobases, the largest among known RNA viruses”. The “basis” units on the molecule are nanometers in size] have found a lovely place to self-replicate, and with each replication, there are chances of “errors”, and so it can rapidly find out for itself new ways to reproduce even more times.

The problem is, therefore, “the global village”. Mass consumerism. Mass tourism. Basically, the Earth is one cruise-ship. One busy shopping mall.

I would like to see the graphs in square root scale or even log scale. You will better be able to see the shapes, and you will more easily see that the places where the numbers are small are actually the noisiest, in a relative sense.

Today’s insights

I was going to write about some new insights here. I posted a picture. Then later added more pictures, to which I need a quick URL in order to refer to them elsewhere. So the new insights never came. But I think I still know what they were. Maybe I’ll write about them another time.

1000 points uniformly distributed on the sphere using the Archimedes theorem method (z is uniform [-1, +1]
2 “t-slices” of a uniform random sample (x, y, z, t) from S^3
Empirical historgram, and theoretical density of, t
“t-slices” of a uniform random sample (x, y, z, t) from S^3. N = 100 000. Delta t = 0.01. Red, radius 0.3. Blue, radius 0.8.
Histogram of x-coordinate of sample of ca. 52000 uniformly distributed points in 3-ball. Theoretical density = parabola. Also drawn: density of x-coordinate of uniformly dist points on 3-sphere = semicircle.
(x, y, z) coordinates of sample of size 1000 of uniform random points (x, y, z, t) on the 3-sphere.
1000 (x,y,z) points from uniform random sample of points (x,y,z,t) on 3-sphere.

My first WordPress post

This is my very first post on my very own WordPress site.

I’m Richard (David) Gill, a mathematical statistician in the Netherlands; originally from England. I’m an emeritus professor of Mathematical Statistics in the Faculty of Science, Leiden University, Netherlands (aka “Holland”), Europe. If you google “Richard Gill statistics” you may well get to know more about me. My usual internet name is “gill1109”. Guess why.