Two points are randomly selected on a line of length $1$. What is the probability that one of the segments is greater than $\frac{1}{2}$? Points can be placed anywhere between [0, 1], for example. Thanks!
[Math] Two points are randomly selected on a line of length $1$
probability
Related Solutions
Choosing two independently and uniformly distributed points $x$, $y\in[0,1]$ is the same as choosing an uniformly distributed point $(x,y)\in Q:=[0,1]^2$. In other words: The joint probability measure at stake is just the area measure on $Q$. Consider a point $(x,y)\in Q$. When $y\leq x$ the requirement that none of the three parts of the stick so generated has length $<{1\over4}$ is fulfilled iff $y\geq{1\over 4}$, $\>x\leq{3\over4}$, and $y\leq x-{1\over4}$. The points fulfilling these conditions are the points in the lower right red triangle in the following figure. The assumption $y\geq x$ gives rise to a congruent such triangle, colored red in the figure as well. All in all, the set of points where none of the three pieces of the stick has length $<{1\over4}$ has area ${1\over16}$. It follows that the probability in question is ${1\over16}$.
Does this make sense to you? I think this aligns with what Ethan has commented.
Best Answer
Probability is $3/4$ if points $x$ and $y$ are chosen with uniform probability. That corresponds to the area in color in the picture below.