Help, there’s a cricket on my floor! I want to trap it with a cup so that I can safely move it outside. But every time I get close, it hops exactly 1 foot in a random direction.
I take note of its starting position and come closer. Boom — it hops in a random direction. I get close again. Boom — it takes another hop in a random direction, independent of the direction of the first hop.
What is the most probable distance between the cricket’s current position after two random jumps and its starting position? (Note: This puzzle is not asking for the expected distance, but rather the most probable distance. In other words, if you consider the probability distribution over all possible distances, where is the peak of this distribution?)
Consider the grasshopper's jump. To end up with a distance of $2$ feet from the starting position, the grasshopper must jump again in exactly the same direction. To end up with a distance of $0$ feet, the grasshopper must jump in exactly the opposite direction.
Note that any other distance, $0 < d < 2$, the grasshopper has two possibilities, either on the left or right of its initial jump direction.
The grasshoppers final distance from the starting point is uniformly distributed for $0 < d < 2$. Exactly $0$ and $2$ feet are half as likely than any specific distance from $0 < d < 2$.