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
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This anomaly is indicative of the larger challenge of culturing various microbial species, referred to as microbial “unculturability.” This cannot be explained by the use of agar alone or by the substitution of an alternative gelling agent, but rather by the difficulties in consistently recreating on an agar plate the multi-variable environment in which microbes grow naturally. Given such challenges, the risk of shortages, and the vulnerabilities of the agar supply chain, why is it so difficult to find suitable alternatives?，推荐阅读同城约会获取更多信息