A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on t

I think this might be correct, Mellie   .....look at the following pic.....

Let the "green" value be an x value lying on segment AC. This segment has the equation y = x. Then, let the "purple" value that is twice a selected x value lie on the segment AE. And the equation of this line is y = 2x. Notice at point E, the x value = .5 and the y value = 1. And at every x value on AC  greater than .5, the  y value on EC associated with this x value is is less than twice that x value.

So, between segments AE and AC, every y value will be greated than, but less than twice, its associated x value. Note that triangle ACD has an area of 1/2. And triangle BEA has an area of 1/4. So, triangle AEC - the area  of interest - has an area of 1 - (1/2) - (1/4)   = 1/4.  And this is the probability that a "purple" value will be greater than, but less than twice,  a "green" value on [0, 1 ].

Thanks Chris,     

I wish I had time to do all your great questions Mellie