## self-avoiding random angles

Posted in Books, Kids, pictures, Statistics, Travel with tags , , , , , on June 17, 2022 by xi'an

An apparently easy riddle from The Riddler this week: given N random half-lines starting from a N-S segment, what is the probability that none ever intersect? If m ½-lines are on the same side of the segment, they will not intersect when their angle with the segment is decreasing with the longitude of the endpoint of this ½-line on the segment. Assuming that drawing a ½-line at random is akin to uniformely drawing an angle on (0,π), this no X’ing event happens when the m angles are properly ordered, a 1/m! event. Independently, the probability for the segments on the other side is 1/(N-m)! The joint probability is thus

$\displaystyle\sum_{m=0}^N {N\choose m}\frac{1}{2^N} \frac{1}{m!(N-m)!}=\frac{(2N)!}{2^N(N!)^3}$

## Байкало-Амурская магистраль/БАМ

Posted in Statistics with tags , , , , , , , , , on April 15, 2018 by xi'an

While in Chamonix last month I dropped by the Guérin editions bookstore, always full of tantalising books on climbing and mountaineering, travelling and travellers. I managed to escape with only two small books, one on a young climber stuck for 100 hours at the top of Aiguilles Vertes [not far from my last ice-climb!] and the other one a railway trip along the Baïkal-Amour Mainline (BAM), which goes from the Baïkal Lake to Sovietskaïa Gavan, north of the Trans-Siberian line. The book is not the ultimate travel book as most of the pages are about historical features surrounding this line, first and foremost the constant reminder that Gulag prisoners were relentlessly exploited to build this line, which follows a macabre route along Siberian camps. The trip finishes not at the end of the BAM line or in Vladivostok, but on Sakhaline Island, which was a penitential colony from the mid-1800’s, as covered by Anton Tchekov in a statistical study and a short story, The Murder… (Comments about characters crossed throughout the trip are rarely to the benefit of these characters.) While I do not make this travel book or the places it crosses sound particularly exciting, it still carries with it the inducing whiff of faraway places, which makes me wish I could see Lake Baïkal or Vladivostok one day in the future, if not travel the entire line. And it also brought back memories of Corto Maltese in Siberia, which remains one of my favourites…

## art brut

Posted in pictures, Running with tags , , , on June 29, 2013 by xi'an