Betrand's Paradox

The Bertrand paradox is generally presented as follows:

Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability $p$ that the chord is longer than a side of the triangle?

Three different solutions are presented, each hinging on the method of generating the random chord:

• METHOD A (random endpoints): Choose two random points on the circumference of the circle, and draw the chord joining them.

• METHOD B (random radial points): Choose a random radius of the circle, and a random point on this radius, and draw the chord through this point and perpendicular to the radius.

• METHOD C (random midpoints): Choose a point anywhere within the circle, and construct the chord such that the point chosen is the midpoint of the chord.

The three different methods above, all seemingly valid, yield different results for the probability in question! The exact answer can be worked out using geometric reasoning, but the goal of this assignment is to provide a visual and empirical way of calculating the probabilities.

Solution

Animation of the solution will pop up on an external browser 😃

library(animation);
library(plotrix);


line.length <- function(points){
  return(norm(points[1,]-points[2,], type="2"));
}


draw.chord <- function(points, threshold){
  color <- "blue";
  chord.length <- line.length(points);
  longer <- chord.length > threshold;
  if(longer){
    color <- "red";
  }
  lines(points[,1], points[,2], col=color);
  
  return(longer);
}


draw.chord.by.midpoint <- function(center.x, center.y, radius,
                                   midpoint.x, midpoint.y){
  
  side.length <- sin(pi/3)*radius*2;
  
  len <- sqrt((midpoint.x-center.x)^2+(midpoint.y-center.y)^2);
  theta <- atan2(midpoint.y-center.y, midpoint.x-center.x);
  
  chord.halflength = sqrt(radius^2-len^2);
  theta1 <- theta + pi/2;
  theta2 <- theta1 + pi;
  
  chord.endpoint1.x <- midpoint.x + cos(theta1)*chord.halflength;
  chord.endpoint1.y <- midpoint.y + sin(theta1)*chord.halflength;
  chord.endpoint2.x <- midpoint.x + cos(theta2)*chord.halflength;
  chord.endpoint2.y <- midpoint.y + sin(theta2)*chord.halflength;
  
  points <- matrix(c(chord.endpoint1.x, chord.endpoint2.x,
                     chord.endpoint1.y, chord.endpoint2.y), nrow=2, ncol=2)
  
  return(draw.chord(points, side.length));
}

# METHOD A -----------------------------------------
method.one <- function(center.x, center.y, radius){
  endpoint1.theta <- runif(1,0,2*pi);
  endpoint2.theta <- runif(1,0,2*pi);
  endpoint1.x = center.x + radius * cos(endpoint1.theta);
  endpoint1.y = center.y + radius * sin(endpoint1.theta);
  endpoint2.x = center.x + radius * cos(endpoint2.theta);
  endpoint2.y = center.y + radius * sin(endpoint2.theta);
  midpoint.x = ( endpoint1.x + endpoint2.x )/2;
  midpoint.y = (endpoint1.y + endpoint2.y)/2;
  return(c(midpoint.x,midpoint.y));
}

# METHOD B -----------------------------------------
method.two <- function(center.x, center.y, radius){
  radius.theta = runif(1,0,2*pi);
  d.from.center = runif(1,0,radius);
  midpoint.x = center.x + d.from.center * cos(radius.theta);
  midpoint.y = center.y + d.from.center  * sin(radius.theta);
  
  return(c(midpoint.x,midpoint.y));
}

# METHOD C -----------------------------------------
method.three <- function(center.x, center.y, radius){
  midpoint.x = center.x - radius;
  midpoint.y = center.y - radius;
  while((midpoint.x-center.x) ^ 2 + (midpoint.y-center.y) ^ 2 > radius ^ 2){
    dx.from.edge = runif(1,0,2*radius) ;
    dy.from.edge = runif(1,0,2*radius) ;
    midpoint.x = center.x - radius + dx.from.edge ;
    midpoint.y = center.y - radius + dy.from.edge ;  
  }
  return(c(midpoint.x,midpoint.y));
}

# MAIN---------------------------------
main <- function(){
  n.trials <- 50;
  xrange <- 49;
  yrange <- 49;
  center.x <- 25;
  center.y <- 25;
  radius <- 24;
  hold.frames <- 25;
  
  generate.midpoints <- function(method){
    midpoints <- vector(mode="list", length=n.trials);
    
    for(iteration in 1:n.trials){
      midpoints[[iteration]] <- method(center.x, center.y, radius);
    }
    
    return(midpoints);
  }
  
  draw.frame <- function(midpoints, n.chords){
    longer <- 0;
    total <- 0;
    plot(1:xrange, 1:yrange, type="n", xlab="", ylab="", main="Bertrand", asp=1);
    draw.circle(center.x, center.y, radius);
    for(chord.idx in 1:n.chords){
      total <- total + 1;
      midpoint <- midpoints[[chord.idx]];
      is.longer <- draw.chord.by.midpoint(center.x, center.y, radius,
                                          midpoint[1], midpoint[2]);
      if(is.longer){
        longer <- longer + 1;
      }
    }
    string <- sprintf("Total: %d, Longer: %d, Probability: %f",
                      total, longer, longer/total);
    mtext(string);
  }
  
  ani.options(interval=0.1);
  ms <- c(method.one, method.two, method.three);
  saveHTML({
    for(method in ms){
      midpoints <- generate.midpoints(method);
      for(iteration in 1:n.trials){
        draw.frame(midpoints, iteration);
      }
      for(i in 1:hold.frames){
        draw.frame(midpoints, n.trials);
      }
    }
  });
}


main()

References

Bertrand's Paradox by Amin Asadi