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The same decomposition works in higher dimensions. For NNN random points on a sphere, the probability they all lie in a hemisphere is also N/2N−1N / 2^{N-1}N/2N−1. The argument is identical: anchor a hemisphere at each point, observe that each event has probability 1/2N−11/2^{N-1}1/2N−1, and verify that at most one anchor can work (the complementary cap is at least a full hemisphere, so no other anchor's hemisphere can straddle it). TODO: If I figure out how to add 3d visualizations to this website, I'll cover the 3D case，推荐阅读搜狗输入法下载获取更多信息
The semicircle is not fixed
If you choose to pre-order the new Samsung Galaxy Buds 4 from Amazon, you'll get a free $20 Amazon gift card. The Samsung Galaxy Buds 4 are listed for $179.99 on Amazon. The gift card offer is valid until March 10, the day before the release of these new earbuds. If you were already lining up these brand-new earbuds, why wouldn't you make the most of this free credit?