Category: Good Statistics

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.

Source:


Harry Crane and Glenn Shafer (2020), Risk is random: The magic of the d’Alembert. https://researchers.one/articles/20.08.00007

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

set.seed(12345)
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

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, https://www.rivm.nl/documenten/epidemiologische-situatie-covid-19-in-nederland-2-april-2020

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.