Question 1151395
<pre>
Uniform distribution means the distribution is a rectangle over
the interval [0,a] whose area is 1. The base of the rectangle is
the interval from 0 to a, so the base of the rectangle is "a" units.
Let its height be h.


{{{drawing(400,800/9,-9,9,-1,3,
locate(-5.1,0,0), locate(4.9,0,a), locate(5.1,1.4,h),locate(-5.5,1.4,h),
line(-10,0,10,0), line(-5,0,-5,2), line(-5,2,5,2), line(-5,0,-5,2), line(5,0,5,2)
)}}}

Area = base∙height
   1 = a∙h
 1/a = h

{{{drawing(400,800/9,-9,9,-1,3,
locate(-5.1,0,0), locate(4.9,0,a), locate(5.1,1.7,1/a),locate(-5.5,1.7,1/a),
line(-10,0,10,0), line(-5,0,-5,2), line(-5,2,5,2), line(-5,0,-5,2), line(5,0,5,2)
)}}}
  
We need to locate a place on that graph where

{{{x>x^2}}}
{{{x^2<x}}}
{{{x^2-x<0}}}
{{{x(x-1)<0}}}

Since x is positive, x-1 must be negative,

{{{x-1<0}}}
{{{x<1}}}

So we locate 1 somewhere on that graph:

{{{drawing(400,800/9,-9,9,-1,3,locate(-5.5,1.7,1/a),
locate(-5.1,0,0), locate(4.9,0,a), locate(5.1,1.7,1/a),
line(-10,0,10,0), line(-5,0,-5,2), line(-5,2,5,2), line(-5,0,-5,2), line(5,0,5,2),line(2,0,2,2), locate(1.9,0,1)
)}}}

and we shade the area under the rectangle to the left of 1, since it's x < 1:

{{{drawing(400,800/9,-9,9,-1,3,locate(-5.5,1.7,1/a),
locate(-5.1,0,0), locate(4.9,0,a), locate(5.1,1.7,1/a),
line(-10,0,10,0), line(-5,0,-5,2), line(-5,2,5,2), line(-5,0,-5,2), line(5,0,5,2),line(2,0,2,2), locate(1.9,0,1),


line(-5,0,-5,2),line(-4.95,0,-4.95,2),line(-4.9,0,-4.9,2),line(-4.85,0,-4.85,2),line(-4.8,0,-4.8,2),line(-4.75,0,-4.75,2),line(-4.7,0,-4.7,2),line(-4.65,0,-4.65,2),line(-4.6,0,-4.6,2),line(-4.55,0,-4.55,2),line(-4.5,0,-4.5,2),line(-4.45,0,-4.45,2),line(-4.4,0,-4.4,2),line(-4.35,0,-4.35,2),line(-4.3,0,-4.3,2),line(-4.25,0,-4.25,2),line(-4.2,0,-4.2,2),line(-4.15,0,-4.15,2),line(-4.1,0,-4.1,2),line(-4.05,0,-4.05,2),line(-4.0,0,-4.0,2),line(-3.95,0,-3.95,2),line(-3.9,0,-3.9,2),line(-3.85,0,-3.85,2),line(-3.8,0,-3.8,2),line(-3.75,0,-3.75,2),line(-3.7,0,-3.7,2),line(-3.65,0,-3.65,2),line(-3.6,0,-3.6,2),line(-3.55,0,-3.55,2),line(-3.5,0,-3.5,2),line(-3.45,0,-3.45,2),line(-3.4,0,-3.4,2),line(-3.35,0,-3.35,2),line(-3.3,0,-3.3,2),line(-3.25,0,-3.25,2),line(-3.2,0,-3.2,2),line(-3.15,0,-3.15,2),line(-3.1,0,-3.1,2),line(-3.05,0,-3.05,2),line(-3.0,0,-3.0,2),line(-2.95,0,-2.95,2),line(-2.9,0,-2.9,2),line(-2.85,0,-2.85,2),line(-2.8,0,-2.8,2),line(-2.75,0,-2.75,2),line(-2.7,0,-2.7,2),line(-2.65,0,-2.65,2),line(-2.6,0,-2.6,2),line(-2.55,0,-2.55,2),line(-2.5,0,-2.5,2),line(-2.45,0,-2.45,2),line(-2.4,0,-2.4,2),line(-2.35,0,-2.35,2),line(-2.3,0,-2.3,2),line(-2.25,0,-2.25,2),line(-2.2,0,-2.2,2),line(-2.15,0,-2.15,2),line(-2.1,0,-2.1,2),line(-2.05,0,-2.05,2),line(-2.0,0,-2.0,2),line(-1.95,0,-1.95,2),line(-1.9,0,-1.9,2),line(-1.85,0,-1.85,2),line(-1.8,0,-1.8,2),line(-1.75,0,-1.75,2),line(-1.7,0,-1.7,2),line(-1.65,0,-1.65,2),line(-1.6,0,-1.6,2),line(-1.55,0,-1.55,2),line(-1.5,0,-1.5,2),line(-1.45,0,-1.45,2),line(-1.4,0,-1.4,2),line(-1.35,0,-1.35,2),line(-1.3,0,-1.3,2),line(-1.25,0,-1.25,2),line(-1.2,0,-1.2,2),line(-1.15,0,-1.15,2),line(-1.1,0,-1.1,2),line(-1.05,0,-1.05,2),line(-1.0,0,-1.0,2),line(-0.95,0,-0.95,2),line(-0.9,0,-0.9,2),line(-0.85,0,-0.85,2),line(-0.8,0,-0.8,2),line(-0.75,0,-0.75,2),line(-0.7,0,-0.7,2),line(-0.65,0,-0.65,2),line(-0.6,0,-0.6,2),line(-0.55,0,-0.55,2),line(-0.5,0,-0.5,2),line(-0.45,0,-0.45,2),line(-0.4,0,-0.4,2),line(-0.35,0,-0.35,2),line(-0.3,0,-0.3,2),line(-0.25,0,-0.25,2),line(-0.2,0,-0.2,2),line(-0.15,0,-0.15,2),line(-0.1,0,-0.1,2),line(-0.05,0,-0.05,2),line(0.0000000000000093952623,0,0.0000000000000093952623,2),line(0.05,0,0.05,2),line(0.1,0,0.1,2),line(0.15,0,0.15,2),line(0.2,0,0.2,2),line(0.25,0,0.25,2),line(0.3,0,0.3,2),line(0.35,0,0.35,2),line(0.4,0,0.4,2),line(0.45,0,0.45,2),line(0.5,0,0.5,2),line(0.55,0,0.55,2),line(0.6,0,0.6,2),line(0.65,0,0.65,2),line(0.7,0,0.7,2),line(0.75,0,0.75,2),line(0.8,0,0.8,2),line(0.85,0,0.85,2),line(0.9,0,0.9,2),line(0.95,0,0.95,2),line(1.0,0,1.0,2),line(1.05,0,1.05,2),line(1.1,0,1.1,2),line(1.15,0,1.15,2),line(1.2,0,1.2,2),line(1.25,0,1.25,2),line(1.3,0,1.3,2),line(1.35,0,1.35,2),line(1.4,0,1.4,2),line(1.45,0,1.45,2),line(1.5,0,1.5,2),line(1.55,0,1.55,2),line(1.6,0,1.6,2),line(1.65,0,1.65,2),line(1.7,0,1.7,2),line(1.75,0,1.75,2),line(1.8,0,1.8,2),line(1.85,0,1.85,2),line(1.9,0,1.9,2),line(1.95,0,1.95,2),line(2.0,0,2.0,2)



)}}}

The area of that shaded part, which has base 1 and height 1/a is

Area = base × height = 1 × 1/a = 1/a

That's the answer: the probability is 1/a.

Edwin</pre>