I have a rectangular section of constant height and length and I choose a random starting point anywhere along its length. From the randomly chose starting point, I first add x units of red material immediately to its right and then fill the remaining section to the right of the red material with blue material.
Here's a graphic I drew up as an example:
My questions is, how do I go about finding the probability that the variable blue material section will be greater than the static red material section?
And also is there a way to determine an average size of the blue section?
My idea was that that the average starting point would be in the middle of the rectangle, therefore the average blue material section would be:
$$ \frac{length\_rectangle}{2} - x $$
Is that correct?
For the probability component, I figure that for the red material section to be greater than the blue section, the starting point would have to start at such a point that: $$ (length\_rectagle - start\_point) < 2x $$ Therefore, the probability that the starting point satisfies the inequality, would be just less than: $$ \frac{2x}{length\_rectangle} $$
The random variable Y (starting point) is uniformly distributed. The condition is, that the blue area is greater than the red area. It is like this, if the random variable $Y$ is smaller than $l-2x$.
l=length of the rectangle
$$P(Y < l-2x)=\frac{l-2x-0}{l-0}=1-\frac{2x}{l}$$