Question
Let’s say we have a number line from $0$ to $7200$, and point $A$ on that line. What is the probability of a random point $B$ being picked within $20$ of point $A$?
My Approach
I imagined splitting the number line into regions $\alpha$, $\beta$, and $\Gamma$, each being assigned to the intervals $[0,20]$, $[20,7180]$, and $[7180,7200]$, respectively. I want to do this because I know that I would just add up all the probabilities for the same question but for $A$ in regions $\alpha$, $\beta$, and $\Gamma$ (since those regions together make up the line $[0,7200]$).
I know that the probability of $A$ being in region $\beta$ while $B$ ends up being $20$ of $A$ should be $\frac{20+20}{7200}=\frac{1}{180}$, but I’m really unsure of how to find the probabilities of this same situation but in the other regions $\alpha$ and $\Gamma$, because if $A$ is in those regions then there is never $20$ of space on both sides of $A$ at the same time.
I’m not sure if it helps, but I was able to reason that in $\alpha$, you can express the probability of $B$ being within $20$ of a fixed $A$ with the function
$$P(A)=\frac{A+20}{7200}$$
Would you find the average of this function for the probability for $\alpha$? And if so would you do something similar for $\Gamma$?