# Size Doesn’t Matter

An invisible red thread connects those destined to meet, regardless of time, place or circumstances. The thread may stretch or tangle, but never break (Ancient Chinese Legend)

I use to play once a year with my friends to Secret Santa (in Spain we call it Amigo Invisible). As you can read in Wikipedia:

To decide who gives whom, every year is the same: one of us introduces small papers in a bag with the names of participants (one name per paper). Then, each of us picks one paper and sees the name privately. If no one picks their own name,  the distribution is valid. If not, we have to start over. Every year we have to repeat process several times until obtaining a valid distribution. Why? Because we are victims of The Matching Problem.

Following the spirit of this talk I have done 16 simulations of the matching problem (for 10, 20, 30 … to 160 items). For example, given n items, I generate 5.000 random vectors sampling without replacement the set of natural numbers from 1 to n. Comparing these random vectors with the ordered one (1,2, …, n) I obtain number of matchings (that is, number of times where ith element of the random vector is equal to i). This is the result of the experiment:

In spite of each of one represents a different number of matchings, all plots are extremely similar. All of them say that probability of not matching any two identical items is around 36% (look at the first bar of all of them). In concrete terms, this probability tends to `1/e` (=36,8%) as n increases but does it very quickly.

This result is shocking. It means that if some day the 7 billion people of the world agree to play Secret Santa all together (how nice it would be!), the probability that at least one person chooses his/her own name is around 2/3. Absolutely amazing.

This is the code (note: all lines except two are for plotting):

```library(ggplot2)
library(scales)
library(RColorBrewer)
library(gridExtra)
library(extrafont)
results=data.frame(size=numeric(0), x=numeric(0))
for (i in seq(10, by=10, length.out = 16)){results=rbind(results, data.frame(size=i, x=replicate(5000, {sum(seq(1:i)-sample(seq(1:i), size=i, replace=FALSE)==0)})))}
opts=theme(
panel.background = element_rect(fill="gray98"),
panel.border = element_rect(colour="black", fill=NA),
axis.line = element_line(size = 0.5, colour = "black"),
axis.ticks = element_line(colour="black"),
panel.grid.major.y = element_line(colour="gray80"),
panel.grid.major.x = element_blank(),
panel.grid.minor = element_blank(),
axis.text.y = element_text(colour="gray25", size=15),
axis.text.x = element_text(colour="gray25", size=15),
text = element_text(family="Humor Sans", size=15, colour="gray25"),
legend.key = element_blank(),
legend.position = "none",
legend.background = element_blank(),
plot.title = element_text(size = 18))
sizes=unique(results\$size)
for (i in 1:length(sizes))
{
data=subset(results, size==sizes[i])
assign(paste("g", i, sep=""),
ggplot(data, aes(x=as.factor(x), weight=1/nrow(data)))+
geom_bar(binwidth=.5, fill=sample(brewer.pal(9,"Set1"), 1), alpha=.85, colour="gray50")+
scale_y_continuous(limits=c(0,.4), expand = c(0, 0), "Probability", labels = percent)+
scale_x_discrete(limit =as.factor(0:8), expand = c(0, 0), "Number of matches")+
labs(title = paste("Matching", as.character(sizes[i]), "items ...", sep=" "))+
opts)
}
grid.arrange(g1, g2, g3, g4, g5, g6, g7, g8, g9, g10, g11, g12, g13, g14, g15, g16, ncol=4)
```

# Shakespeare Is More Monkey-Friendly Than Cervantes

Ford, there is an infinite number of monkeys outside who want to talk to us about this script for Hamlet they have worked out (from Episode 2 of The Hitchhiker’s Guide to the Galaxy by Douglas Adams)

Some days ago I was talking with a friend about the infinite monkey theorem which is a funny interpretation of what thinking-in-infinite can produce. The same day, in my weekly English class, my teacher said that Anglo-saxon words do tend to be short, very often monosyllabic such as function words such as to, of, from etc and everyday words such as go, see run, eat, etc.

Both things made me think that a monkey could have easier to type a Shakespeare text rather than a Cervantes one. I cannot imagine a definitive way to demonstrate this but this experiment support my hypothesis. After simulating random words of 2, 3, 4 and 5 characters I look for them in English(1) and Spanish(2) dictionaries, which I previously downloaded from here. Result: I find more random words in the English one. These are the results of my experiment:

For example, around 38% of two-chars words match with English dictionary and only 9% with Spanish one. This is why I think that, in the infinite, I would be easier for a monkey to replicate a Shakespeare text than a Cervantes one.

Here you have the code:

```library(ggplot2)
library(scales)
df.lang=do.call("rbind", list(esp.dic, eng.dic))
df.lang\$WORD=tolower(iconv(df.lang\$WORD, to="ASCII//TRANSLIT"))
df.lang=unique(df.lang)
results=data.frame(LANG=character(0), OCCURRENCES=numeric(0), SIZE=numeric(0), LENGTH=numeric(0))
for (i in 2:5)
{
df.monkey=data.frame(WORD=replicate(20000, paste(sample(c(letters), i, replace = TRUE), collapse='')))
results=rbind(results, data.frame(setNames(aggregate(WORD ~ ., data = merge(df.lang, df.monkey, by="WORD"), FUN=length), c("LANG","OCCURRENCES")), SIZE=20000, LENGTH=i))
}
opt=theme(panel.background = element_rect(fill="gray92"),
panel.grid.minor = element_blank(),
panel.grid.major.x = element_blank(),
panel.grid.major.y = element_line(color="white", size=1.5),
plot.title = element_text(size = 35),
axis.title = element_text(size = 20, color="gray35"),
axis.text = element_text(size=16),
axis.ticks = element_blank(),
axis.line = element_line(colour = "white"))
ggplot(data=results, aes(x=LENGTH, y=OCCURRENCES/SIZE, colour=LANG))+
geom_line(size = 2)+
scale_colour_discrete(guide = FALSE) +
geom_point(aes(fill=LANG),size=10, colour="gray92",pch=21)+
scale_x_continuous("word length", labels=c("two chars", "three chars", "four chars", "five chars"))+
scale_y_continuous("probability of existence", limits=c(0, 0.4), labels = percent)+
labs(title = "What if you put a monkey in front of a typewriter?")+
opt + scale_fill_discrete(name="Dictionary", breaks=c("ESP", "ENG"), labels=c("Spanish", "English"))
```

(1) The English dictionary was originally compiled from public domain sources
for the amSpell spell-checker by Erik Frambach e-mail: e.h.m.frambach@eco.rug.nl
(2) The Spanish dictionary has been elaborated by Juan L. Varona, Dpto. de Matematicas y Computacion, Universidad de La Rioja, Calle Luis de Ulloa s/n, 26004 SPAIN e-mail: jvarona@siur.unirioja.es