## Le Monde puzzle [#1024]

Posted in Books, Kids with tags , , , , , , , on October 10, 2017 by xi'an The penultimate and appropriately somewhat Monty Hallesque Le Monde mathematical puzzle of the competition!

A dresser with 5×5 drawers contains a single object in one of the 25 drawers. A player opens a drawer at random and, after each choice, the object moves at random to a drawer adjacent to its current location and the drawer chosen by the player remains open. What is the maximum number of drawers one need to open to find the object?

In a dresser with 9 drawers in a line, containing again a single object, the player opens drawers one at a time, after which the open drawer is closed and the object moves to one of the drawers adjacent to its current location. What is the maximum number of drawers one need to open to find the object?

For the first question, setting a pattern of exploration and, given this pattern, simulating a random walk trying to avoid the said pattern as long as possible is feasible, returning a maximum number of steps over many random walks [and hence a lower bound on the true maximum]. As in the following code

```sefavyd=function(pater=seq(1,49,2)%%25+1){
fild=matrix(0,5,5)
m=pater;i=fild[m]=1
t=sample((1:25)[-m],1)
nomove=FALSE
while (!nomove){
i=i+1
m=pater[i];fild[m]=1
if (t==m){ nomove=TRUE}else{
muv=NULL
if ((t-1)%%5>0) muv=c(muv,t-1)
if (t%%5>0) muv=c(muv,t+1)
if ((t-1)%/%5>0) muv=c(muv,t-5)
if (t%/%5<4) muv=c(muv,t+5)
muv=muv[fild[muv]==0]
nomove=(length(muv)==0)
if (!nomove) t=sample(rep(muv,2),1)}
}
return(i)}
```

But a direct reasoning starts from the observation that, while two adjacent drawers are not opened, a random walk can, with non-zero probability, switch indefinitely between both drawers. Hence, a sure recovery of the object requires opening one drawer out of two. The minimal number of drawers to open on a 5×5 dresser is 2+3+2+3+2=12. Since in 12 steps, those drawers are all open, spotting the object may require up to 13 steps.

For the second case, unless I [again!] misread the question, whatever pattern one picks for the exploration, there is always a non-zero probability to avoid discovery after an arbitrary number of steps. The [wrong!] answer is thus infinity. To cross-check this reasoning, I wrote the following R code that mimics a random pattern of exploration, associated by an opportunistic random walk that avoids discovery whenever possible (even with very low probability) bu pushing the object towards the centre,

```drawl=function(){
i=1;t=5;nomove=FALSE
m=sample((1:9)[-t],1)
while (!nomove){
nextm=sample((1:9),1)
muv=c(t-1,t+1)
muv=muv[(muv>0)&(muv<10)&(muv!=nextm)]
nomove=(length(muv)==0)||(i>1e6)
if (!nomove) t=sample(rep(muv,2),1,
prob=1/(5.5-rep(muv,2))^4)
i=i+1}
return(i)}
```

which returns unlimited values on repeated runs. However, I was wrong and the R code unable to dismiss my a priori!, as later discussions with Robin and Julien at Paris-Dauphine exhibited ways of terminating the random walk in 18, then 15, then 14 steps! The idea was to push the target to one of the endpoints because it would then have no option but turning back: an opening pattern like 2, 3, 4, 5, 6, 7, 8, 8 would take care of a hidden object starting in an even drawer, while the following 7, 6, 5, 4, 3, 2 openings would terminate any random path starting from an odd drawer. To double check:

```grawl=function(){
len=0;muvz=c(3:8,8:1)
for (t in 1:9){
i=1;m=muvz[i];nomove=(t==m)
while (!nomove){
i=i+1;m=muvz[i];muv=c(t-1,t+1)
muv=muv[(muv>0)&(muv<10)&(muv!=m)]
nomove=(length(muv)==0)
if (!nomove)
t=sample(rep(muv,2),1)}
len=max(len,i)}
return(len)}
```

produces the value 14.

## Monty Hall closes the door

Posted in Books, Kids, pictures with tags , , , , , , , , , on October 1, 2017 by xi'an Among much more dramatic news today, I learned about Monty Hall passing away, who achieved long lasting fame among probabilists for his TV game show leading to the Monty Hall problem, a simple conditional probability derivation often leading to arguments because of the loose wording of the conditioning event. By virtue of Stigler’s Law, the Monty Hall game was actually invented earlier, apparently by the French probabilist Joseph Bertrand, in his Calcul des probabilités. The New York Times article linked with the image points out the role of outfits with the game participants, towards being selected by the host, Monty Hall. And that one show had a live elephant behind a door, instead of a goat, elephant which freaked out..!

## Measuring statistical evidence using relative belief [book review]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , , , , on July 22, 2015 by xi'an

“It is necessary to be vigilant to ensure that attempts to be mathematically general do not lead us to introduce absurdities into discussions of inference.” (p.8) This new book by Michael Evans (Toronto) summarises his views on statistical evidence (expanded in a large number of papers), which are a quite unique mix of Bayesian  principles and less-Bayesian methodologies. I am quite glad I could receive a version of the book before it was published by CRC Press, thanks to Rob Carver (and Keith O’Rourke for warning me about it). [Warning: this is a rather long review and post, so readers may chose to opt out now!]

“The Bayes factor does not behave appropriately as a measure of belief, but it does behave appropriately as a measure of evidence.” (p.87)

## The Monty Hall “problem”

Posted in Books, Statistics with tags , , , on February 4, 2010 by xi'an

I stumbled by chance on this book The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser on Amazon, or rather and more accurately Amazon suggested the book as connected to Burdzy’s The Search for Certainty. I first thought why would anyone need a whole book for explaining a simple conditioning argument (and the fallacy of conditioning on the wrong event) that I usually give as a problem to my second year undergraduates. But then I started reading the comments and found one that could not believe there was such a book because the answer was clearly 50-50! (Obviously, this comment was written by someone who had not read the book…) And I thus vaguely remembered a story about a highly respectable and respected statistician getting trapped by this puzzle… So maybe a book is in order. Maybe. But I find the argument of one of the commenters of the above disbelieving comment quite convincing: imagine there are 10,000 doors (instead of just 3), you pick one, the host opens 9,998 out of the 9,999 remaining ones and let you decide between switching  to the last remaining door and sticking to your original choice. Would you ever stick?!