Doing Math with Python_ Bertrand's Paradox(1)

What's the Problem?

Above is an equilateral triange and its circumcircle. Let's say the length of a side of the triangle is 'a', and the circumcircle is called 'O'. In this case, what would the percentage of a random chord in circle O's length exceeding 'a'? 

Expected Answers

1) 1/3

 

2) 1/4

 

Using Python

Script 1 _ Random X Coordinate Point

import random
import math
import turtle as t
 
# variables
xa=0
ya=0
xb=0
yb=0
d=0
y=0
big=[]
i=0
j=0
r1=200
r2=1
f=[]
c='none'
tf=0
final=0
 
#axis
t.speed(0)
t.penup()
t.goto(-400,-300)
t.pendown()
t.goto(400,-300)
t.penup()
t.goto(-400,-300)
t.pendown()
t.goto(-400,300)
 
# program
t.hideturtle()
t.penup()
t.goto(200,100)
t.pendown()
t.circle(100)
while j<r2:
    while i<r1:
        xa=random.random()
        y=[math.sqrt((1/4)-(xa-1/2)**2),-(math.sqrt((1/4)-(xa-1/2)**2))]
        ya=random.choice(y)
        xb=random.random()
        y=[math.sqrt((1/4)-(xb-1/2)**2),-(math.sqrt((1/4)-(xb-1/2)**2))]
        yb=random.choice(y)
 
        d=math.sqrt((xa-xb)**2+(ya-yb)**2)
        if d>=math.sqrt(3/4):
            big=big+[d]
            c='red'
            tf=1
        else:
            c='black'
            tf=0
 
        i=i+1
 
        #circle graphics
        t.penup()
        t.goto(200*(xa-1/2)+200,200*(ya)+200)
        t.pendown()
        t.color(c)
        t.goto(200*(xb-1/2)+200,200*(yb)+200)
 
        #graph
        t.penup()
        t.goto((800/r1)*(i-1)-400,600*((len(big)-tf)/i)-300)
        t.pendown()
        t.goto((800/r1)*(i)-400,600*(len(big)/(i+1))-300)
        
 
    f=f+[(len(big)/r1)*100]
    j=j+1
 
final=float(sum(f)/len(f))
t.write(final,False,"right",("",15))
print(final)
 

<Result>

Script 2 _ Random Angle Point

 

import random
import math
import turtle as t
 
#variables
radian=0
xa=0
ya=0
xb=0
yb=0
big=[]
c='none'
r1=200
r2=1
tf=0
i=0
j=0
final=[]
result=0
 
#grid
t.speed(0)
t.penup()
t.goto(-400,-300)
t.pendown()
t.goto(400,-300)
t.penup()
t.goto(-400,-300)
t.pendown()
t.goto(-400,300)
 
#program
t.hideturtle()
t.penup()
t.goto(200,100)
t.pendown()
t.circle(100)
while j<r2:
    while i<r1:
        radian=2*(random.random())*(math.pi)
        xa=1*math.cos(radian)
        ya=1*math.sin(radian)
        radian=2*(random.random())*(math.pi)
        xb=1*math.cos(radian)
        yb=1*math.sin(radian)
 
        d=math.sqrt((xa-xb)**2+(ya-yb)**2)
        if d>=math.sqrt(3):
            big=big+[d]
            c='red'
            tf=1
        else:
            c='black'
            tf=0
 
        i=i+1
 
        #circle graphic
        t.penup()
        t.goto(100*xa+200,100*ya+200)
        t.pendown()
        t.color(c)
        t.goto(100*xb+200,100*yb+200)
 
        #graph
        t.penup()
        t.goto((800/r1)*(i-1)-400,600*((len(big)-tf)/i)-300)
        t.pendown()
        t.goto((800/r1)*(i)-400,600*(len(big)/(i+1))-300)
 
    final=final+[(len(big)/r1)*100]
    j=j+1
 
result=float(sum(final)/len(final))
t.write(result,False,"right",("",15))
print(result)