Visualization of Bertrand paradox

bertrand-paradox.png

bertrand-paradox.R

library(ggplot2)

bertrand <- list(
  function(n) {
    theta1 <- runif(n, 0, 2*pi)
    theta2 <- runif(n, 0, 2*pi)
    data.frame(x1 = cos(theta1), x2 = cos(theta2),
               y1 = sin(theta1), y2 = sin(theta2))
  },
  function(n) {
    theta <- runif(n, 0, 2*pi)
    r <- runif(n, 0, 1)
    data.frame(x1 = cos(theta + acos(r)), x2 = cos(theta - acos(r)),
               y1 = sin(theta + acos(r)), y2 = sin(theta - acos(r)))
  },
  function(n) {
    theta <- runif(n, 0, 2*pi)
    r <- sqrt(runif(n, 0, 1))
    data.frame(x1 = cos(theta + acos(r)), x2 = cos(theta - acos(r)),
               y1 = sin(theta + acos(r)), y2 = sin(theta - acos(r)))
  }
)

df <- lapply(bertrand, do.call, list(n=1e3)) |> 
  dplyr::bind_rows(.id = "id") |> 
  dplyr::mutate(length = sqrt((x1-x2)^2 + (y1-y2)^2),
                larger = length > 2*sin(pi/3))

df |> dplyr::summarise(mean(length), mean(larger), .by=id)

p1 <- ggplot(df) + facet_grid(~id) +
  labs(title = "Distribution of chords") +
  aes(x = x1, xend = x2, y = y1, yend = y2, color = larger) +
  coord_fixed() + xlim(-1, 1) + ylim(-1, 1) + theme_void() +
  scale_color_manual(values = c("gray11", "red")) +
  geom_segment(size = 0.05, show.legend = FALSE)

p2 <- ggplot(df) + facet_grid(~id) +
  labs(title = "Distribution of midpoints") +
  aes(x = (x1+x2)/2, y = (y1+y2)/2, color = larger) +
  coord_fixed() + xlim(-1, 1) + ylim(-1, 1) + theme_void() +
  scale_color_manual(values = c("gray11", "red")) +
  geom_point(size = 0.05, show.legend = FALSE)

p3 <- ggplot(df) + facet_grid(~id) +
  labs(title = "Distribution of lengths") +
  aes(x = length, fill = larger) +
  theme_linedraw() +
  scale_fill_manual(values = c("gray11", "red")) +
  geom_histogram(show.legend = FALSE)

patchwork::wrap_plots(p1, p2, p3, ncol=1)