## Friday, September 16, 2011

### Three Darts in One Hemisphere

You throw three darts onto the surface of a globe, each from a randomly chosen direction. What is the probability that all three darts are in one hemisphere?

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100% on the upper hemisphere

ReplyDelete1/3?

ReplyDeleteThe probability is 1. You can draw a circle through any three points in space, and since any section of a sphere leaves a circular cut the circle through three darts must lie on the surface of the sphere. Since this circle could not be bigger than a great circle, they must lie in a single hemisphere.

ReplyDeleteConfused? What is the answer?

ReplyDeleteAidan and Anonymous have it right. The probability is 1 that all three darts are in one hemisphere.

ReplyDeleteFor a longer explanation than what Aidan gives, you can check out this elementary probability book: http://portal.tpu.ru/SHARED/k/KITAEVA/statistics/book/Tab2/ElementaryProbabilityDavidStirzaker.pdf

Just search through the document for the keyword sphere. Then you'll have to read over the chapter.

Warning, it's not that elementary to figure out the math!