Question

Two points are selected randomly on a line of length 32 so as to be on opposite sides of the midpoint of the line. In other words, the two points X and Y are independent random variables such that X is uniformly distributed over [0,16) and Y is uniformly distributed over (16,32]. Find the probability that the distance between the two points is greater than 9.

Answer 1

Let us consider the squares be
`$[1, 16] * [16, 32]$`

If x ranges from the 0 to 16 and the y ranges from 16 to 32, we see that the boundary of the region
`$y-x \geq 9 \text{\ is}\ y - x = 9$` which goes from the
`$(7, 16 ) \text{ to}\ (16, 25) $`.

And so it is easier to find the area of region where
`$ y-x \leq 9$`. This is the triangle with points
`$(7,16),(16,25) \text{ and}\ (16,16)$` as its vertices.

The area if the triangle is =
`$(1)/(2) * 9 * 9$`

=
`$(81)/(2)$`

Now the entire area is
`$(16)^2$` = 256

Then,
`$P(y-x \leq 9) = (81/2)/(256) =(81)/(512)$`

or
`$P(y-x \geq 9) = 1 - (81)/(512)=(431)/(512)$`

Thus the answer is
`$(431)/(512)$`