riddle by attrition

The weekend riddle from The Riddler is rather straightforward [my wording and simplification]:

Construct a decimal number X between 0 and 1 by drawing the first digit a¹ uniformly over {0,1,…,9}, the second digit a² uniformly over {0,1,…,9}, &tc., until 0 is attained. What is the expectation of this random variable X?

Since each new digit has expectation half of the previous digit, the expectation is an infinite geometric series with common ratio 20⁻¹ and factor 9, leading to an expectation of 9/19. As verified with the following R code:

skr<-function(){
  a=9;b=0
  while((a<-sample(rep(0:a,2),1))>0)b=10*b+a
  while(b>=1)b=b/10
  return(b)}

3 Responses to “riddle by attrition”

  1. […] December 2, 2019 By Donald Greer [This article was first published on R – Xi’an’s Og, and kindly contributed to R-bloggers]. (You can report issue about the content on this page […]

  2. […] article was first published on R – Xi'an's Og, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here) […]

  3. […] by data_admin [This article was first published on R – Xi’an’s Og, and kindly contributed to R-bloggers]. (You can report issue about the content on this page […]

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