Bertand’s paradox [R details]

March 19, 2011

(This article was first published on Xi'an's Og » R, and kindly contributed to R-bloggers)

Some may have had reservations about the “randomness” of the straws I plotted to illustrate Bertrand’s paradox. As they were all going North-West/South-East. I had actually made an inversion between cbind and rbind in the R code, which explained for this non-random orientation. Above is the corrected version, which sounds “more random” indeed. (And using wheat as the proper, if weak, colour!) The outcome of a probability of 1/2 has not changed, of course. Here is the R code as well:


for (t in 1:10^3){

 #distance from O to chord

 while (dchord>1){
 #Generate "random" straw in large box till it crosses unit circle


 #endpoints outside the circle
 if ((sum(a^2)>1)&&(sum(b^2)>1)){


 #chord inside
 if (max(abs(theta),abs(thetb))<pi/2)

 if (runif(1)<.1) lines(rbind(a,b),col="wheat")


As a more relevant final remark, I came to the conclusion (this morning while running) that the probability of this event can be anything between 0 and 1, rather than the three traditional 1/4, 1/3 and 1/2. Indeed, for any distribution of the “random” straws, hence for any distribution on the chord length L, a random draw can be expressed as L=F⁻¹(U), where U is uniform. Therefore, this draw is also an acceptable transform of a uniform draw, just like Bertrand’s three solutions.

Filed under: Books, R, Statistics Tagged: Bertrand’s paradox, chord, E.T. Jaynes, height, probability theory, R, simulation, triangle

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