1×1=2
You are told that, when rounded to the nearest whole number, x and y are both 1. What is the probability that, to the nearest whole number, xy = 2 ?
Solution
Assuming that both x and y are both from continuous uniform distributions with and 0.5
y 1.5
then 0.5
 xy

1.5
when both x and y
0.5
x 1.5
are both in the yellow part of the square:
The blue area is

1
1/ 2
1
2 x d x
1 ln 2

1
4 2 4
.
The red area is
3
4

3/ 2

1
3
2 x
The yellow area is 1
 d x

3ln
2
3 
3
2 4 ln 2

1

2 4



.
3ln
2
3 
3
2 4





 ln
27
16 
1
2 2
or 0.762
to 3 d.p.
1 of 2
TB v1.0 © MEI
31/07/2015

Paper folding a parabola
Prove that the following steps will produce a parabola:
 Fold a piece of A4 paper in half long-ways (to create the y-axis) and draw this in with a pen.
 Mark a point on the fold a couple of inches from the bottom edge.
 Fold the bottom edge so that it goes through the point and is perpendicular to the vertical fold (to create the x-axis) and mark this with a pen.
 Make repeated folds so that the bottom edge goes through the point – these can be at any angle.
Solution
A series of points, A, are folded onto the point P.
Each fold is a perpendicular bisector and contributes a point to the envelope of folds that is directly above the point A which has been folded onto P.
By writing the point P as (0, ) and the general point on the curve B as coordinates
( , ) then A has
( ,
 a ) and hence:
AB = PB y
  x
2   y )
2 y
2 
2 ay
 a
2  x
2  a
2 
2 ay
 y
2
4 ay
 x
2 x
2 y

4 a
Which is a parabola.
2 of 2
TB v1.0 © MEI
31/07/2015