I'm posting one puzzle, riddle, math, or statistical problem a day. Try to answer each one and post your answers in the comments section. I'll post the answer the next day. Even if you have the same answer as someone else, feel free to put up your answer, too!
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?
The 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.
Aidan and Anonymous have it right. The probability is 1 that all three darts are in one hemisphere.
For 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!
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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!