Archive for waiting time

piano tuners on France Inter

Posted in Books, Kids, University life with tags , , , , , , , , , , , on August 14, 2026 by xi'an

During the summer months, the Belgian physicist Fabrizio Bucella delivers a daily and funny unit of wisdom every morning just before 8am (CET) on the French public radio. Often addressing every day question to uncover a scientific principle. A recent morn, he mentioned Fermi’s estimates, which is a numerate way to guess(timate) a physical quantity by multiplying individual estimates, like the number of piano tuners in Chicago. (My preparatory school physics teacher was very good at this game, beating our pocket calculators by a large margin!) But I had never heard either of the concept or of the expression. Bucella’s 2mn explanation was quite similar to the Wikipedia entry, if not mentioning that the squared root increase of the error applies to the logarithm of the estimate. And arguing unreasonably at all terms being above or below their true values with probability ½. The paradox of the waiting time the day after was more convincing, if failing to mention Siméon Poisson. [The above is a picture I took of Lake Michigan while running, the last time I visited Chicago. A few days before Barack Obama got re-elected.]

uniform spacings

Posted in Books, Kids, R with tags , , , , on June 8, 2023 by xi'an

A riddle on uniform spacings!, namely when considering eight iid Uniform (0,1) variates as visiting times and three further iid Uniform (0,1) variates as server availability times, with unit service time, the question being the probability a server is available for a ninth visiting time, T⁹. Which can be decomposed into four cases:

  1. at least one server becomes available between the previous visiting time and T⁹
  2. at least two servers become available between the penultimate visiting time and the previous visiting time
  3. two servers become available between the antepenultimate visiting time and penultimate visiting time and one server becomes available between the penultimate visiting time and the previous visiting time
  4. three servers become available between the antepenultimate visiting time and the penultimate visiting time

with respective probabilities (using Devroye’s Theorem 2.1)

1/4, 8 9!3!/12!, 7 8!3!/12!, 7 8!3!/12!,

resulting in a total probability of 0.2934, compatible with a simulation assessment.