From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Sent: Sunday, March 4, 2012 6:45 PM Subject: [math-fun] Identity (5)
Consider the standard binary tree with infinitely many levels.
Suppose each edge is colored green with probability = p.
What is the probability f(p) that there exists an infinite green path starting at the root?
--Dan
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f(p) = (2p - 1)/p^2 for p > 1/2, f(p) = 0 for p <= 1/2.
-- Gene
Yow, slope 8 at p=1/2+ ! Maybe building a tree is a good way to test coins for fairness. --rwg
A coin can be characterized by a quantity q, depending on its geometry, internal density distribution, and the prescribed tossing process, which is experimentally manifested as the probability of heads. When a coin has been repeatedly tossed, resulting in h heads and t tails, there is no better way to estimate q than by the standard application of Bayes' theorem. Before the coin is tossed, our state of knowledge is given by some prior probability p0(q), and after the tosses, our updated knowledge is given by the posterior probability p(q | h, t) = N p0(q) q^h (1 - q)^t. The normalization constant N is chosen to make int(p(q | h, t), q = 0..1) = 1. If we take the prior to be uniform, p(q | h, t) = ((h + t + 1)! / (h! t!)) q^h (1 - q)^t. -- Gene