Stop the alien invasion!

Today’s puzzle is from The Riddler, and has to do with spherical geometry.

A guardian constantly patrols a spherical planet, protecting it from alien invaders that threaten its very existence. One fateful day, the sirens blare: A pair of hostile aliens have landed at two random locations on the surface of the planet. Each has one piece of a weapon that, if combined with the other piece, will destroy the planet instantly. The two aliens race to meet each other at their midpoint on the surface to assemble the weapon. The guardian, who begins at another random location on the surface, detects the landing positions of both intruders. If she reaches them before they meet, she can stop the attack.

The aliens move at the same speed as one another. What is the probability that the guardian saves the planet if her linear speed is 20 times that of the aliens’?

Here is my solution for the case of interest, where the guardian is faster than the intruders.
[Show Solution]

Here is a partial solution for the more complicated case where the guardian is slower than the intruders.
[Show Solution]

3 thoughts on “Stop the alien invasion!”

  1. Since the guardian’s speed is given as 20 times that of the aliens’ speed, should it not be 10 times the combined speed of the two aliens as they are moving towards each other at the same speed?

    1. I interpreted the question to mean that Alien A and Alien B each have some speed $s$, and the guardian has speed $20s$. You can think of this as the distance between both aliens shrinking with speed $2s$, or you can think of it as each alien moving toward point $M$ with speed $s$.

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