Can You Estimate The Distance?¶

Riddler Express ¶

Amare the ant is back! This week, he’s trying to escape from a spider web.

Starting at the very center of the web, he crawls 1 inch radially away in a random direction. From there, he randomly chooses between crawling in the radial or tangential directions, with a 50 percent chance of each. When he crawls radially, he moves another inch from his current position directly away from the center of the web. When he crawls tangentially, he moves another inch in one of the two directions perpendicular to the radial direction. (To be clear, when Amare moves tangentially, he is still moving farther away from the center of the web, at least a little bit.)

After Amare has crawled a total of 4 inches, how far away from the center of the web should he expect to be on average?

Solution¶

I started with a point 1 away from the center, (0,1). This was the sole stop in the first stops. The weight, or probability of this stop is 1. I ran some code:

Starting with the first stop in the previous list of stops, I calculated the slope from the center to that stop. I also calculated the tangential slope.
Using the stop and the tangential slope, the two tangential child stops were calculated and given a weight of $\dfrac{1}{4}$ of the parent stop.
Using the stop and the slope, the radial child stop was calculated and given a weight of $\dfrac{1}{2}$ of the parent stop.
These three new child stops were appended to the new stops.
This process was repeated until every parent stop in the previous stops spawned three new child stops in the new stops.
I repeated this process until I had 10 lists of stops.

Answer¶

The following animation follows all possible stops Amare could end up at. The area of the stop corresponds with the weight or probability.

After Amare has traveled 4 inches, he is on average 2.95 inches away from the center.

Rohan Lewis¶

2022.05.23¶