Olly, Olly, Olly, you gave me a bad headache.I thought the...

Olly, Olly, Olly, you gave me a bad headache.

I thought the answer was 50%, BUT I was wrong.

My logic went.

throw the 4 darts, A,B,C and D.

draw a circle around A, B and C. Now push this circle away (on a parallel plane) until it becomes a great circle. Now there is a 50% chance D will land on the same side of this great circle as A, B and C.... easy... WRONG.

I did some googling and found,

http://www.mathpages.com/home/kmath327/kmath327.htm

he/she did Monte Carlo simulation and got 87.5%. (i.e. write computer code to do the same dart throwing process, do it a lot of times and find the probability of all being in the same hemisphere)

%%%%%%%

Im having a lot of trouble figuring out why the answer is not 1-0.5^4 = 0.9375.

i.e. Either D can be inside the projected great circle of ABC, or C inside the circle ABD.... and so on....