c*w
发帖数: 4736 | 1 First, the probability that they are on one semisphere is 1-prob(not
on
one semisphere). Now I'll study the prob that they are NOT on one
semisphere. However, the first n are always on one semisphere, then
if the last one make them NOT in one semisphere, then its "Fan3 Xiang4
Yan2 Chang2 Xian4 will be within the wierd part that "cut" by those
planes that are formed by (n-1) points and the center, all together
there
are n such planes, and cut the whole ball into a small Jiao3. So as
far
as the ar | C******a
发帖数: 115 | 2 我这样考虑，以球面为例，其上的四个点何时处在一个半球面上。
如果它们在一个半球面上，则存在一个点和这四个点的内积非负。
问题变为讨论此点的存在性。以上述四点为心的四个半球面分别
是与这些点内积非负的点集。如果它们有交集，那四个点就在一
个半球面上。因此可以考虑任意四个半球面有交集的概率有多大。
事实上每个半球面是由一个大圆生成，而一个大圆同时生成两个
半球面。对每个大圆生成的半球面做不同选择，总共考虑2^4=16
次不同的情形。而四个半球面相交的次数是一个随机变量，其期
望是所求概率的16倍。忽略概率零的情形，假设任意三个大圆不
交于一点。半球面相交的次数正好是四个大圆把球面分成的块数。
如果任意三个大圆不交于一点，则四个大圆将球面分成十四部分。
所以概率是14/16=7/8=1-1/8。
呵呵，我考虑的时间太长了，不好意思。
【在 c*w 的大作中提到】
: First, the probability that they are on one semisphere is 1-prob(not
: on
: one semisphere). Now I'll study the prob that they are NOT on one
: semisphere. However, the first n are always on one semisphere, then
: if the last one make them NOT in one semisphere, then its "Fan3 Xiang4
: Yan2 Chang2 Xian4 will be within the wierd part that "cut" by those
: planes that are formed by (n-1) points and the center, all together
: there
: are n such planes, and cut the whole ball into a small Jiao3. So as
: far
| c*w
发帖数: 4736 | 3 I perfectly agree with you that the walls may not be
independent, and therefore the area of each cone are
dependent of eachother. However, the expected value
of the area of each cone must be the same, because
the area is determined by their vertices only, which
follow the same distribution.
And with basic knowledge of probability, we know that
E(a1 + a2 + ..... + am) = E(a1) +E(a2) + ... E(am)
(however dependent a1....am are!!!!).
However, a1 + ... + am = whole area of the sphere,
and plus that |
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