Goats do room

The riddle of the week is about 10 goats sequentially moving to their room, which they have chosen at random and independently (among ten rooms), unless another goat already occupies the room, in which case they move to the first free room with a higher number or fail. What is the probability that all goats end up in a room?

Coding the experiment is straightforward:

g=sample(1:N,N,rep=TRUE)
o=0*g
for(i in 1:N){
    if(min(o[g[i]:N])){f=f+1;break()
    }else{
      o[min(which(!o[g[i]:N]))+g[i]-1]=1
    }}}

returning an estimated probability of approximately 0.764.

As I had some free time during the early mornings at ISBA 2022, I tried to reformulate the question as a continuous event on uniform order statistics, turning to be at most one uniform larger than (N-1)/N, at most two larger than (N-2)/N, and so on… Asking the question on math.stackexchange quickly produced an answer that reversed engineered my formulation back to the goats (or parking lot), with a generic probability of

which of course coincides with the Monte Carlo approximation!

As an aside, I once drank South-African wines named Goats-do-Roam and Goat-Roti at my friends Jim and Maria’s place,  and they were quite enjoyable!

Beaulort (Sauternes)

internal workshop in Montpellier

Today was a meeting day for our research (ANR) network EMILE and I flew to Montpellier in the early morning, barely catching my 7am flight by a mere 8 minutes, thanks to a huge unannounced gap (more than 30mn!) in the distribution of the metro trains… Anyway, it was a very nice day with interesting talks on on-going researchs by several members of the network, including a new type of (non-ABC) approximation for phylogenetic trees, INLA on genotype distribution, Bayesian tree estimation with SNP data, and the new version of the DIYABC software. (Jean-Michel Marin and I also presented our recent work on ABC model choice and advertised the incoming Read Paper on ABC methods to the group, as they could contribute to the discussion.) One of the talks involved the pseudo-Bayes factors (CPO) of Geisser and Eddy discussed recently in connection with the book reviews of both Bayesian ideas and data analysis and Bayesian modeling using WinBUGS. Unfortunately, again estimated by an harmonic mean…