The probability of it NOT happening goes like this...
Area of a sphere: A = 4 * pi * R^2
Area of a cap: a = 2 * pi * R * h {where h is the height of the cap}
Let's use a unit sphere: R = 1
So: A = 4 * pi a = 2 * pi * h
So probability for each cap is: p = h / 2
Observe that the area is linear in "h", and that the probability is linear in "h".
So the overall probability is: P = (p[h = 0] + p[h = 1]) / 2 P = ((0 / 2) + (1 / 2)) / 2 P = 1 / 4
The math there is kind of tough sledding, and one of my principles is that one doesn't understand anyting unless one can explain it to an eight year old so:
Their proof goes like this...
Let's call the points A, B, C, and D.
Any two points define a great circle.
Take the circle defined by A + B, pick the hemisphere C is in, then it's 50% that D is also in it.
Take the circle defined by A + C, pick the hemisphere B is in, then it's 50% that D is also in it.
Take the circle defined by B + C, pick the hemisphere A is in, then it's 50% that D is also in it.
Probability that D was in none of them = (1 / 2)^3 = 1 / 8
Makes the answer 87.5%
I have no friggin idea so I googled it. I hope you math guru's appreciated my cut and paste because I still have no friggin idea.
