So a problem I've been struggling with for a while is the following
Assume you are placed within the unit circle at a unspecified location $P$, and a predator is placed somewhere on on the circumference of the circle. As your velocity on land is far greater than the predator, you escape if you manage to reach the shore first. However, while in water the predators velocity is 4 times greater than yours. What is the probability you are able to escape, assuming you are just as likely to be at any point within the circle.
Naive attempt
So, I simplified the problem due to rotation symmetry and drew the following
With $L$ being the location of the predator and $G$ being our random position. Switching to polar coordinates gives
$$\color{blue}{\text{Arc = }} \pi - 2 \theta, \qquad \color{red}{\|GE\| = } 2 \cos \theta - \rho$$
Since the red segment must be 4 times greater than the blue arc we obtain
$$ \rho > 2 \cos \theta - \frac{\pi - 2\theta}{4}$$
In order to escape. For different velocities it can look like this
Where you are captured no matter what if you are within the colored circle. Finding the probability of not getting captured is now simply
$$P = 1 - \frac{1}{\pi/2} \int_0^{\pi/2} \frac{1}{2}\bigl(2 \cos \theta - \frac{\pi - 2\theta}{4}\bigr)^2\,\mathrm{d}\theta = \frac{2}{\pi} - \frac{\pi^2}{96} \approx 53\%$$
Issue
My solution assumes that we run in the diametrically opposite direction.
However, looking at the following picture it is not clear this is optimal
Maybe following the normal ($m$) gives a better result, than diametrically opposite ($i$), or perhaps the optimal path is somewhere between ($2$).
Questions
- Assuming you are randomly placed within the unit circle, what is the optimal path to get to the shore faster than your predator.
- Assuming the predator runs 4 times as fast, what is the chance of you escaping? (Again assuming your placement is uniformly distributed within the unit circle)