A point $A$, is chosen at random with uniform distribution on a line segment of length $L$. A second point, $B$, is chosen randomly, independently of the first. What is the probability that $B$ is closer to $A$ than to either of the endpoints of the segment?
I'm not sure how to go about this problem. I've set it up so the distance from the the beginning (call this point 0) to $B$ can be expressed as $B$, the distance from $L$ to $B$ can be expressed as $L-B$, and the distance between A and B is $|A-B|$, but I'm unsure of how to find the desired probability.
Best Answer
Simplify the problem. Let the line segment be $[0,1]$ and suppose that $A=0.5$. Given this fixed $A$, what is the probability for a random $B$?
Now try it again for $A=0.6$ and $A=0.2$. Does the answer change? If so, how? If not, why not?