Computing the probability that the rounding of y/x be an even number, where x, y are uniformly distributed on [0,1].

rounding_division.rkt

#lang racket
(require plot)

(define (ct p)
  (match-define (vector x y) p)
  (/ y x))

(define (f g)
  (λ (p)
    (zero? (modulo (inexact->exact (g (ct p))) 2))))

(define p (/ (- 5 pi) 4))

(define (ps N)
  (for/list ([k (in-range N)])
    (vector (random) (random))))

; do
; > (run)
; to run an experiment
(define (run [N (expt 10 6)])
  (define g round)
  (define margin 100)
  (define p*
    (/
     (length (filter (f g) (ps N)))  N))
  (define ϵ (* margin (abs (- p* p))))
  (displayln (/ ϵ margin))
  (chebyshev ϵ N))

(define (chebyshev ϵ m)
  (- 1 (/ (* p (- 1 p)) (* m ϵ ϵ))))

; > (run-plot)
; plot the points of a run
(define (run-plot [N (expt 10 6)])
  (plot
   (points
    (filter (f round) (ps N))
    #:sym 'dot)))