@MISC{Richardson06np, author = {Ross M. Richardson}, title = {n p}, year = {2006} }

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Abstract

These notes accompany a lecture given in the summer of 2006 at the Center for Combinatorics at Nankai University. They are provided as a reference (and especially bibliography) for students new to sharp concentration phenomena. 1 A Basic Problem Perhaps one of the most basic examples in probability theory is that of the coin flip sequence. We flip a biased coin n times which is heads with probability p, and count the number of occurrences of heads, X. Clearly, X is binomially distributed with parameters n and p. While it is possible that X = n, it is elementary that this probability is vanishingly small, p n. The probability that X = n or n − 1 is and in general P(X ≥ k) = p n + np n−1 (1 − p), n∑ i=k