A Visit from our Neighbours

Discrete Mathematics Level 4

Some time in the near future (about \(300\) million years from now), our neighbours from the planet \(\text{Alpha Centauri Bb}\) decide to visit Earth. Since they have no idea about the places of interest on Earth and there are no humans alive (on Earth) to guide them either, they decide to pick a random longitude and latitude and visit that place. The aliens will be thrilled iff the chosen coordinate is underwater.

Let the probability that the aliens are thrilled be \(P\). Find \(\lfloor 100 P \rfloor\).

Details and assumptions:

  • Assume that Earth is a perfect sphere, and \(70.8 \%\) of Earth's surface is underwater.
  • The longitude and latitude are chosen randomly and uniformly from \((-180 ^{\circ} , 180 ^{\circ} ]\) and \([- 90 ^{\circ}, 90 ^{\circ} ]\) respectively.

  • Check out a follow-up tougher problem in my solution once you have attempted this problem (or revealed solution).

Image credit: ESO/L. Cal├žada/Nick Risinger (skysurvey.org)

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